Alternating series test – Serlo

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An introductory video to the alternating series test. (German) (YouTube-Video , channel: Quatematik)

The alternating series criterion serves to prove convergence of an alternating series, i.e. a series where the pre-signs alternately change from positive to negative, like ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ or ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}b_{k}}$ (with all ${\displaystyle b_{k}}$ being positive). Series of this kind can be convergent, but not absolutely convergent. I those cases, criteria for absolute convergence will fail, but the alternating series criterion may be successful.

The alternating series criterion goes back to Gottfried Wilhelm Leibniz and was published in 1682.

Introductory example: Convergence of the alternating harmonic series

Series treated by the alternating series criterion will often converge, but not converge absolutely. Perhaps, the most prominent example for such a converging, but not absolutely convergent series is the alternating harmonic series ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}}$. Convergence of it can be shown by making sure that the sequence of partial sums ${\displaystyle (S_{n})_{n\in \mathbb {N} }=\left(\sum _{k=1}^{n}{\frac {(-1)^{k+1}}{k}}\right)_{n\in \mathbb {N} }}$ converges. For ${\displaystyle n=1,2,3,4,5,6,7,8}$ , those partial sums are

The partial sums - visualized
{\displaystyle {\begin{aligned}S_{1}&=\sum _{k=1}^{1}{\frac {(-1)^{k+1}}{k}}=1,&S_{2}&=\sum _{k=1}^{2}{\frac {(-1)^{k+1}}{k}}={\frac {1}{2}}=0.5,\\S_{3}&=\sum _{k=1}^{3}{\frac {(-1)^{k+1}}{k}}={\frac {5}{6}}=0.8{\overline {3}},&S_{4}&=\sum _{k=1}^{4}{\frac {(-1)^{k+1}}{k}}={\frac {7}{12}}=0.58{\overline {3}},\\S_{5}&=\sum _{k=1}^{5}{\frac {(-1)^{k+1}}{k}}={\frac {47}{60}}=0.78{\overline {3}},&S_{6}&=\sum _{k=1}^{6}{\frac {(-1)^{k+1}}{k}}={\frac {37}{60}}=0.61{\overline {6}},\\S_{7}&=\sum _{k=1}^{7}{\frac {(-1)^{k+1}}{k}}={\frac {329}{420}}=0.759,&S_{8}&=\sum _{k=1}^{8}{\frac {(-1)^{k+1}}{k}}={\frac {533}{840}}=0.634.\end{aligned}}}

Those partial sums tend to make jumps of ever smaller getting distance. Those partial sums with odd indices, (${\displaystyle S_{2n-1}}$) seem to be monotonously decreasing and those with even indices ${\displaystyle S_{2n}}$ seem to be increasing. A simple calculation can mathematically verify this assertion: For all ${\displaystyle n\in \mathbb {N} }$ there is

{\displaystyle {\begin{aligned}S_{2n+1}-S_{2n-1}&=\sum _{k=1}^{2n+1}{\frac {(-1)^{k+1}}{k}}-\sum _{k=1}^{2n-1}{\frac {(-1)^{k+1}}{k}}\\[0.5em]&=({\color {Teal}1-{\frac {1}{2}}+{\frac {1}{3}}-\ldots +{\frac {(-1)^{2n}}{2n-1}}}+{\frac {(-1)^{2n+1}}{2n}}+{\frac {(-1)^{2n+2}}{2n+1}})\\[0.5em]&-({\color {Indigo}1-{\frac {1}{2}}+{\frac {1}{3}}-\ldots +{\frac {(-1)^{2n}}{2n-1}}})\\[0.5em]&={\frac {(-1)^{2n+1}}{2n}}+{\frac {(-1)^{2n+2}}{2n+1}}\\[0.5em]&={\frac {1}{2n+1}}-{\frac {1}{2n}}\\[0.5em]&{\color {OliveGreen}\left\downarrow \ 2n+1\geq 2n\Rightarrow {\frac {1}{2n+1}}\leq {\frac {1}{2n}}\right.}\\[0.5em]&\leq 0,\end{aligned}}}

i.e. ${\displaystyle S_{2n+1}\leq S_{2n-1}}$ (monotonously decreasing). Analogously, ${\displaystyle S_{2n+2}-S_{2n}={\frac {(-1)^{2n+3}}{2n+2}}+{\frac {(-1)^{2n+2}}{2n+1}}=-{\frac {1}{2n+2}}+{\frac {1}{2n+1}}\geq 0}$, so ${\displaystyle S_{2n+2}\geq S_{2n}}$ (monotonously increasing).

If we could now show that ${\displaystyle (S_{2n-1})}$ is bounded from below and ${\displaystyle (S_{2n})}$is bounded from above, then both sequences would converge by the monotony criterion. Luckily, this is exacly the case: all odd partial sums are bounded from below by any even partial sum and all even partial sums are bounded from above by any odd partial sum: For all ${\displaystyle n\in \mathbb {N} }$

${\displaystyle S_{2n-1}-S_{2n}=-{\frac {(-1)^{2n+1}}{2n}}={\frac {(-1)^{2n+2}}{2n}}={\frac {1}{2n}}\geq 0,}$

so ${\displaystyle S_{2n-1}\geq S_{2n}}$ and ${\displaystyle S_{2n}\leq S_{2n-1}}$. We therefore have the bounds ${\displaystyle S_{2n-1}\geq S_{2n}\geq S_{2}={\frac {1}{2}}}$ and ${\displaystyle S_{2n}\leq S_{2n-1}\leq S_{1}=1}$. Hence, ${\displaystyle (S_{2n-1})}$ is bounded from below by ${\displaystyle {\frac {1}{2}}}$ and ${\displaystyle (S_{2n})}$ is bounded from above by ${\displaystyle 1}$ .

The monotony criterion now implies that both the subsequences of partial sums ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ are convergent.

In order to get convergence of the series, we need to show that the sequence of partial sums ${\displaystyle (S_{n})}$ converges. This is for sure the case, if both the odd and the even subsequence ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ converge to the same limit.

How can this be shown? First, let us assign a name too the limits: ${\displaystyle \lim _{n\to \infty }S_{2n-1}=S}$ and ${\displaystyle \lim _{n\to \infty }S_{2n}=S'}$. The statement we want to show can then mathematically be expressed as ${\displaystyle S=S'}$. We show this by subtracting both limits from each other, which is equivalent to taking the limit of the sequence difference:

${\displaystyle S-S'=\lim _{n\to \infty }S_{2n-1}-\lim _{n\to \infty }S_{2n}=\lim _{n\to \infty }(S_{2n-1}-S_{2n}).}$

Above, we showed ${\displaystyle S_{2n-1}-S_{2n}={\frac {1}{2n}}}$ which is a null sequence:

${\displaystyle \lim _{n\to \infty }(S_{2n-1}-S_{2n})=\lim _{n\to \infty }{\frac {1}{2n}}=0.}$

Hence, ${\displaystyle S-S'=0}$ which means ${\displaystyle S=S'}$.

From this, we can imply that the sequence of partial sums ${\displaystyle (S_{n})}$ converges to ${\displaystyle S}$ . Mathematically, we need to stay closer than any ${\displaystyle \epsilon }$ to the limit value after surpassing some sequence element number ${\displaystyle N}$ for the corresponding ${\displaystyle \epsilon }$

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall n\geq N:|S_{n}-S|<\epsilon .}$

For a fixed ${\displaystyle \epsilon }$, both the odd and the even partial sum sequences have a suitable ${\displaystyle N}$, which we name by ${\displaystyle (2N_{1}-1)}$ (odd) and ${\displaystyle 2N_{2}}$(even):

{\displaystyle {\begin{aligned}&\forall \epsilon >0\,\exists N_{1}\in \mathbb {N} \,\forall n\geq N_{1}:|S_{2n-1}-S|<\epsilon \\[0.3em]&\forall \epsilon >0\,\exists N_{2}\in \mathbb {N} \,\forall n\geq N_{2}:|S_{2n}-S|<\epsilon ,\end{aligned}}}

After reaching the greater of these two numbers ${\displaystyle N=\max\{(2N_{1}-1),2N_{2}\}}$, both sequences stay closer than ${\displaystyle \epsilon }$ to the limit value and

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall n\geq N:|S_{n}-S|<\epsilon .}$

Generalizing the proof idea / alternating series test

Now we consider any alternating series. Can we use the same proof as for the alternating harmonic series to show that our general alternating series converges? The answer will depend on the properties of the general alternating series. We used the following properties from the alternating harmonics series:

• The sequence of coefficients ${\displaystyle (b_{k})=\left({\frac {1}{k}}\right)}$ without the alternating presign is monotonously decreasing. This gave us monotony and boundedness of the two partial sums ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ , so we could show that they converge. Without the monotony, this may not be the case.
• Further, we used that ${\displaystyle (b_{k})=\left({\frac {1}{k}}\right)}$ is a null sequence. This was needed to show that both ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ had the same limit, so ${\displaystyle (S_{n})}$ converges to that limit. If ${\displaystyle (b_{k})=\left({\frac {1}{k}}\right)}$ converged to a constant ${\displaystyle B}$, then ${\displaystyle (S_{n})}$ would in the end tend to "jump" up and down by an amount of ${\displaystyle B}$ and the limits of ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ may differ by ${\displaystyle B}$, so they are not equal.

No further properties of the alternating harmonic series have been used for the proof. So we may use the above proof steps to show convergence of a general alternating series:

Theorem (Alternating series test)

Let ${\displaystyle (b_{k})_{k\in \mathbb {N} }}$ be a non-negative, monotonously decreasing null sequence of real numbers, i.e. ${\displaystyle \lim _{k\to \infty }b_{k}=0}$. Then, the alternating series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges.

The proof uses the same steps as the convergence proof for the alternating harmonic series above.

Proof (Alternating series test)

We need to show that the sequence of partial sums ${\displaystyle (S_{n})=\left(\sum _{k=1}^{n}(-1)^{k+1}b_{k}\right)}$ converges.

Step 1: The odd subsequence ${\displaystyle (S_{2n-1})}$ is monotonously decreasing and the even subsequence ${\displaystyle (S_{2n})}$ is monotonously increasing, as for any ${\displaystyle n\in \mathbb {N} }$ there is

{\displaystyle {\begin{aligned}S_{2n+1}-S_{2n-1}&=\sum _{k=1}^{2n+1}(-1)^{k+1}b_{k}-\sum _{k=1}^{2n-1}(-1)^{k+1}b_{k}\\[0.5em]&=({\color {Teal}b_{1}-b_{2}+b_{3}-\ldots +(-1)^{2n}b_{2n-1}}\\[0.5em]&+(-1)^{2n+1}b_{2n}+(-1)^{2n+2}b_{2n+1})\\[0.5em]&-({\color {Indigo}b_{1}-b_{2}+b_{3}-\ldots +(-1)^{2n}b_{2n-1}})\\[0.5em]&=(-1)^{2n+2}b_{2n+1}+(-1)^{2n+1}b_{2n}\\[0.5em]&=b_{2n+1}-b_{2n}\\[0.5em]&{\color {OliveGreen}\left\downarrow \ b_{2n+1}\leq b_{2n}\right.}\\[0.5em]&\leq 0,\end{aligned}}}

and analogously ${\displaystyle S_{2n+2}-S_{2n}=(-1)^{2n+3}b_{2n+2}+(-1)^{2n+2}b_{2n+1}=-b_{2n+2}+b_{2n+1}\geq 0}$.

Step 2: ${\displaystyle (S_{2n-1})}$ is bounded from below and ${\displaystyle (S_{2n})}$ is bounded from above, since for ${\displaystyle n\in \mathbb {N} }$ there is

${\displaystyle S_{2n-1}-S_{2n}=(-1)^{2n+2}b_{2n}=b_{2n}\geq 0}$

So ${\displaystyle S_{2n-1}\geq S_{2n}\geq S_{2}=b_{1}-b_{2}}$ and ${\displaystyle S_{2n}\leq S_{2n-1}\leq S_{1}=b_{1}}$

The monotony criterion yields convergence of both the partial sums ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$.

Step 3: ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ converge to the same limit. Let ${\displaystyle \lim _{n\to \infty }S_{2n-1}=S}$ and ${\displaystyle \lim _{n\to \infty }S_{2n}=S'}$. In step 2, we proved convergence of both sequences, so we can use the sum rule for limits of sequences:

${\displaystyle S-S'=\lim _{n\to \infty }S_{2n-1}-\lim _{n\to \infty }S_{2n}=\lim _{n\to \infty }(S_{2n-1}-S_{2n})}$

On the other hand,

${\displaystyle \lim _{n\to \infty }(S_{2n-1}-S_{2n})=\lim _{n\to \infty }b_{2n}=0,}$

since both ${\displaystyle (b_{k})}$ and ${\displaystyle (b_{2n})}$ are null sequences. So both limits are equal (${\displaystyle S=S'}$).

Step 4: ${\displaystyle (S_{n})}$ also converges to ${\displaystyle S}$. Since ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$ converge to ${\displaystyle S}$ , both approach ${\displaystyle S}$ up to any ${\displaystyle \epsilon }$:

{\displaystyle {\begin{aligned}&\forall \epsilon >0\,\exists N_{1}\in \mathbb {N} \,\forall n\geq N_{1}:|S_{2n-1}-S|<\epsilon \\[0.5em]&\forall \epsilon >0\,\exists N_{2}\in \mathbb {N} \,\forall n\geq N_{2}:|S_{2n}-S|<\epsilon \end{aligned}}}

We now take the greater number ${\displaystyle N=\max\{2N_{1}-1,2N_{2}\}}$ and obtain that also ${\displaystyle (S_{n})}$ approaches ${\displaystyle S}$ up to ${\displaystyle \epsilon }$:

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall n\geq N:|S_{n}-S|<\epsilon }$

So the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges - and we are done with the proof.

Proof alternative

Alternatively, one may use the Cauchy criterion for proving the alternating series criterion.

Alternative proof (Alternating series test)

In order to apply the alternating series criterion, we need to show that (under the assumptions of the alternating series test), there is

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall n\geq m\geq N:\left|\sum _{k=m}^{n}(-1)^{k+1}b_{k}\right|<\epsilon .}$

At first, let us take only odd ${\displaystyle m}$ and estimate the sum ${\displaystyle \sum _{k=m}^{n}(-1)^{k+1}b_{k}}$ . This sum is positive:

{\displaystyle {\begin{aligned}\sum _{k=m}^{n}(-1)^{k+1}b_{k}&=(-1)^{m+1}b_{m}+(-1)^{m+2}b_{m+1}+(-1)^{m+3}b_{m+2}+(-1)^{m+4}b_{m+3}+\ldots \\[0.5em]&\left\downarrow \ m{\text{ odd}}\Rightarrow m+1{\text{ even}}\Rightarrow m+2{\text{ odd}}\right.\\[0.5em]&=b_{m}-b_{m+1}+b_{m+2}-b_{m+3}+\ldots +{\begin{cases}b_{n}&{\text{ if }}n{\text{ odd,}}\\(-b_{n})&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\left\downarrow \ {\text{ associativity}}\right.\\[0.5em]&=(b_{m}-b_{m+1})+(b_{m+2}-b_{m+3})+\ldots +{\begin{cases}b_{n}&{\text{ if }}n{\text{ odd,}}\\(b_{n-1}-b_{n})&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\left\downarrow \ (b_{k}){\text{ monotonously decreasing }}\Rightarrow b_{k}-b_{k+1}\geq 0\right.\\[0.5em]&=\underbrace {(b_{m}-b_{m+1})} _{\geq 0}+\underbrace {(b_{m+2}-b_{m+3})} _{\geq 0}+\ldots +{\begin{cases}\underbrace {b_{n}} _{\geq 0}&{\text{ if }}n{\text{ odd,}}\\\underbrace {(b_{n-1}-b_{n})} _{\geq 0}&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\geq 0.\end{aligned}}}

But it is also bounded:

{\displaystyle {\begin{aligned}\sum _{k=m}^{n}(-1)^{k+1}b_{k}&=(-1)^{m+1}b_{m}+(-1)^{m+2}b_{m+1}+(-1)^{m+3}b_{m+2}+(-1)^{m+4}b_{m+3}+\ldots \\[0.5em]&\left\downarrow \ m{\text{ odd}}\Rightarrow m+1{\text{ even}}\Rightarrow m+2{\text{ odd }}\right.\\[0.5em]&=b_{m}-b_{m+1}+b_{m+2}-b_{m+3}+\ldots +{\begin{cases}b_{n}&{\text{ if }}n{\text{ odd,}}\\(-b_{n})&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\left\downarrow \ {\text{ associativity}}\right.\\[0.5em]&=b_{m}-(b_{m+1}-(b_{m+2})-(b_{m+3}-b_{m+4})+\ldots +{\begin{cases}-(b_{n-1}-b_{n})&{\text{ if }}n{\text{ odd,}}\\-b_{n}&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\left\downarrow \ (b_{k}){\text{ monotonously decreasing }}\Rightarrow b_{k}-b_{k+1}\geq 0\right.\\[0.5em]&=b_{m}\underbrace {-(b_{m+1}-b_{m+2})} _{\leq 0}\underbrace {-(b_{m+3}-b_{m+4})} _{\leq 0}+\ldots +{\begin{cases}\underbrace {-(b_{n-1}-b_{n})} _{\leq 0}&{\text{ if }}n{\text{ odd,}}\\\underbrace {-b_{n}} _{\leq 0}&{\text{ if }}n{\text{ even}}\end{cases}}\\[0.5em]&\leq b_{m}.\end{aligned}}}

So, for odd ${\displaystyle m}$, there is ${\displaystyle \left|\sum _{k=m}^{n}(-1)^{k+1}b_{k}\right|=\sum _{k=m}^{n}(-1)^{k+1}b_{k}\leq b_{m}=|b_{m}|}$.

Analogously, for even ${\displaystyle m}$ , the sum is negative and bounded:

${\displaystyle \sum _{k=m}^{n}(-1)^{k+1}b_{k}\leq 0{\text{ and }}\sum _{k=m}^{n}(-1)^{k+1}b_{k}\leq -b_{m},}$

And the absolute value of the sum is also bounded by ${\displaystyle \left|\sum _{k=m}^{n}(-1)^{k+1}b_{k}\right|\leq |b_{m}|}$ . The latter inequality holds for odd and even, i.e. for all ${\displaystyle m\in \mathbb {N} }$.

But now, ${\displaystyle (b_{k})}$ was assumed to be a null sequence, so ${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall m\geq N:|b_{m}|<\epsilon }$. The inequality for the absolute value hence implies

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall n\geq m\geq N:\left|\sum _{k=m}^{n}(-1)^{k+1}b_{k}\right|<\epsilon .}$

And hence, the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges by the Cauchy criterion.

Application example

Example (Generalized alternating harmonic series)

The generalized alternating harmonic series ${\displaystyle \sum _{k=1}^{\infty }{\tfrac {(-1)^{k+1}}{k^{\alpha }}}}$ converges for all ${\displaystyle \alpha >0}$ by the alternating series test, since

• ${\displaystyle b_{k}={\tfrac {1}{k^{\alpha }}}\geq 0}$ for all ${\displaystyle k\in \mathbb {N} }$.
• ${\displaystyle b_{k+1}={\tfrac {1}{(k+1)^{\alpha }}}{\overset {(k+1)^{\alpha }\geq k^{\alpha }}{\leq }}{\tfrac {1}{k^{\alpha }}}}$ for all ${\displaystyle k\in \mathbb {N} }$, so ${\displaystyle (b_{k})}$ is monotonously decreasing.
• ${\displaystyle \lim _{k\to \infty }b_{k}=\lim _{k\to \infty }{\tfrac {1}{k^{\alpha }}}=0}$, so ${\displaystyle (b_{k})}$ is a null sequence.

Note that for ${\displaystyle \alpha >1}$, this series even converges absolutely, as ${\displaystyle \sum _{k=1}^{\infty }{\tfrac {1}{k^{\alpha }}}=\zeta (\alpha )<\infty }$.

Notes to the alternating series test

• Of course, we can also change the series presigns from positive ${\displaystyle \rightarrow }$ negative ${\displaystyle \rightarrow }$ positive ${\displaystyle \rightarrow ...}$ to negative ${\displaystyle \rightarrow }$ positive ${\displaystyle \rightarrow }$ negative ${\displaystyle \rightarrow ...}$ and get a valid convergence criterion for series like ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}b_{k}}$. The proof is the same, under an interchange of ${\displaystyle (S_{2n-1})}$ and ${\displaystyle (S_{2n})}$.
• One can also start from ${\displaystyle k=0}$, i.e. consider series like ${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}b_{k}}$ or ${\displaystyle \sum _{k=0}^{\infty }(-1)^{k+1}b_{k}}$. Any starting index ${\displaystyle k}$ is OK. The proof is just the same including an index shift.
• As above, the alternating series test does only lead convergence, but no absolute convergence. For instance, the alternating harmonic series ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}}$ converges by the alternating series test. However, it does not converge absolutely.
• The alternating series test can never be used for implying divergence of a series. If a series fails to meet the criteria for the alternating series test, it can still converge. There is an example warning about this below See below.

• The proof for the alternating series implies that ${\displaystyle (I_{n})_{n\in \mathbb {N} }=([c_{n},d_{n}])_{n\in \mathbb {N} }}$ with ${\displaystyle c_{n}=S_{2n}}$ and ${\displaystyle d_{n}=S_{2n-1}}$ is a sequence of nested intervals.
• One can also prove the more general Dirichlet test and then conclude the alternating series test as a special case. Further infos will be given in the end of this article.
• We could also take ${\displaystyle (b_{k})_{k\in \mathbb {N} }}$ to be a non-positive and monotonously increasing null sequence. I.e. it approaches ${\displaystyle 0}$ from below. The proof works the same way. Especially, that means ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges, whenever ${\displaystyle (b_{k})_{k\in \mathbb {N} }}$ is just any monotone null sequence.

A test problem

Exercise (Alternating series test)

Is the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\sqrt {k}}{k+1}}}$ convergent?

How to get to the proof? (Alternating series test)

In order to apply the alternating series test, we need to show that the sequence ${\displaystyle (b_{k})=\left({\frac {\sqrt {k}}{k+1}}\right)}$ is a monotonously decreasing null sequence. That means, we need 2 proof steps:

1. We prove: ${\displaystyle (b_{k})}$ is monotonously decreasing, i.e. ${\displaystyle b_{k+1}\leq b_{k}}$. This can be shown by proving that the ratio between two elements is ${\displaystyle {\frac {b_{k+1}}{b_{k}}}\leq 1}$ or their difference is ${\displaystyle b_{k+1}-b_{k}\leq 0}$ . Differences of square roots are hard to handle, so we investigate the ratio ${\displaystyle {\frac {b_{k+1}}{b_{k}}}\leq 1}$.
2. We prove: ${\displaystyle (b_{k})}$ is a null sequence, i.e. ${\displaystyle \lim _{k\to \infty }b_{k}=0}$. We can prove this using limit theorems for sequences.

Proof (Alternating series test)

The sequence ${\displaystyle (b_{k})=\left({\frac {\sqrt {k}}{k+1}}\right)}$ is nonnegative (${\displaystyle b_{k}\geq 0}$) and

Step 1: ${\displaystyle {\frac {b_{k+1}}{b_{k}}}\leq 1}$

{\displaystyle {\begin{aligned}{\frac {b_{k+1}}{b_{k}}}&={\frac {\frac {\sqrt {k+1}}{k+2}}{\frac {\sqrt {k}}{k+1}}}\\&={\frac {{\sqrt {k+1}}(k+1)}{{\sqrt {k}}(k+2)}}\\[0.5em]&={\frac {\sqrt {(k+1)(k+1)^{2}}}{\sqrt {k(k+2)^{2}}}}\\[0.5em]&={\frac {\sqrt {k^{3}+3k^{2}+3k+1}}{\sqrt {k^{3}+4k^{2}+4k}}}\\[0.5em]&={\sqrt {\frac {k^{3}+3k^{2}+3k+1}{k^{3}+4k^{2}+4k}}}\end{aligned}}}

Now, the square root defines a strictly monotonously increasing function. Hence, it is ${\displaystyle \leq 1}$, whenever the number under the root is ${\displaystyle \leq 1}$. We investigate the behaviour of the expression under the root in the limit ${\displaystyle k\to \infty }$:

{\displaystyle {\begin{aligned}{\frac {k^{3}+3k^{2}+3k+1}{k^{3}+4k^{2}+4k}}&={\frac {k^{3}+3k^{2}+3k+1}{k^{3}+3k^{2}+3k+(k^{2}+k)}}\\[0.5em]&\left\downarrow \ k^{2}+k\geq 1\right.\\[0.5em]&\leq {\frac {k^{3}+3k^{2}+3k+1}{k^{3}+3k^{2}+3k+1}}\\[0.5em]&=1\end{aligned}}}

Hence, ${\displaystyle {\frac {b_{k+1}}{b_{k}}}\leq 1}$, so ${\displaystyle (b_{k})}$ is monotonously decreasing.

Step 2: ${\displaystyle \lim _{k\to \infty }b_{k}=0}$

{\displaystyle {\begin{aligned}\lim _{k\to \infty }b_{k}&=\lim _{k\to \infty }{\frac {\sqrt {k}}{k+1}}\\&\left\downarrow \ {\text{ factorize the denominator }}\right.\\[0.5em]&=\lim _{k\to \infty }{\frac {\sqrt {k}}{{\sqrt {k}}({\sqrt {k}}+{\frac {1}{\sqrt {k}}})}}\\[0.5em]&\left\downarrow \ {\text{cancel }}{\sqrt {k}}\right.\\[0.5em]&=\lim _{k\to \infty }{\frac {1}{{\sqrt {k}}+{\frac {1}{\sqrt {k}}}}}\\[0.5em]&\left\downarrow \ \lim _{k\to \infty }{\frac {1}{\sqrt {k}}}=0{\text{ use limit theorems}}\right.\\[0.5em]&=0\end{aligned}}}

So ${\displaystyle (b_{k})}$ is a null sequence.

Additional question: Does this series converge absolutely?

No, ${\displaystyle \sum _{k=1}^{\infty }\left|(-1)^{k+1}{\frac {\sqrt {k}}{k+1}}\right|=\sum _{k=1}^{\infty }{\frac {\sqrt {k}}{k+1}}}$ diverges. For large ${\displaystyle k}$, the elements scale like ${\displaystyle {\frac {\sqrt {k}}{k+1}}\propto {\frac {1}{\sqrt {k}}}}$, which is even larger than a harmonic series ${\displaystyle {\frac {1}{k}}}$. More precisely,

{\displaystyle {\begin{aligned}{\frac {\sqrt {k}}{k+1}}&\geq {\frac {\sqrt {k}}{k+k}}\\[0.5em]&={\frac {\sqrt {k}}{2k}}\\[0.5em]&={\frac {1}{2{\sqrt {k}}}}\\[0.5em]&\geq {\frac {1}{2k}}\end{aligned}}}

and ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2k}}}$ is a divergent harmonic series.

So we have divergence by direct comparison.

What, if a condition is not fulfilled?

It is important to check that the 2 conditions for the alternating series test are fulfilled! There are alternating series, which do not meet one. The following examples will illustrate alternating series, where ${\displaystyle (b_{k})}$ is either not converging to 0 (our example converges to 1) or not monotonous. Both examples fail to be convergent (although they are alternating). The third example is an alternating series, which fails the alternating series test (as it is not monotonous), but nevertheless converges. So the alternating series test does not identify all convergent alternating series.

Example (Example 1: Convergence to 0 is needed.)

We consider the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}\left({\frac {k+1}{k}}\right)}$ with ${\displaystyle b_{k}={\frac {k+1}{k}}}$. Does it fulfil the requirements for the alternating series test? Let us check:

• ${\displaystyle (b_{k})}$ is monotonously decreasing.

Exercise (Monotony)

Prove it.

Proof (Monotony)

We investigate the ratio ${\displaystyle {\frac {b_{k+1}}{b_{k}}}={\frac {\frac {k+2}{k+1}}{\frac {k+1}{k}}}={\frac {k(k+2)}{(k+1)^{2}}}={\frac {k^{2}+2k}{k^{2}+2k+1}}\leq 1}$ , so ${\displaystyle b_{k+1}\leq b_{k}}$. Hence, ${\displaystyle (b_{k})}$ is monotonously decreasing.

• ${\displaystyle (b_{k})}$ converges to 1, so it is not a null sequence. We can easily see this by explicitly computing the limit: ${\displaystyle \lim _{k\to \infty }b_{k}=\lim _{k\to \infty }{\frac {k+1}{k}}=\lim _{k\to \infty }{\frac {1+{\frac {1}{k}}}{1}}={\frac {1}{1}}=1\neq 0}$.

So the alternating series test can not be applied. And indeed, the series diverges, since for large ${\displaystyle k}$, it essentially behaves like ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}\cdot 1}$ . More precisely, the divergence can be proven by the term test: The sequence of elements ${\displaystyle (a_{k})=\left((-1)^{k}{\frac {k+1}{k}}\right)}$ can not be a null sequence, since the subsequence ${\displaystyle (a_{2n})}$ converges to 1. So the corresponding series diverges by the term test.

Example (Example 2: Monotony is needed.)

Next, we consider the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ with ${\displaystyle b_{k}={\begin{cases}{\frac {1}{\sqrt {k}}}&{\text{ for odd }}k,\\{\frac {1}{k}}&{\text{ for even }}k\end{cases}}}$. Will it converge? Let us try the alternating series test:

• ${\displaystyle (b_{k})}$ is not monotonously decreasing. This can be easily seen considering the first series elements: ${\displaystyle \left(1,{\frac {1}{2}},{\frac {1}{\sqrt {3}}},{\frac {1}{4}},{\frac {1}{\sqrt {5}}},{\frac {1}{6}},{\frac {1}{\sqrt {7}}},{\frac {1}{8}},{\frac {1}{\sqrt {9}}}={\frac {1}{3}},{\frac {1}{10}},\ldots \right)}$. Clearly, ${\displaystyle {\frac {1}{2}}<{\frac {1}{\sqrt {3}}},{\frac {1}{4}}<{\frac {1}{\sqrt {5}}},{\frac {1}{6}}<{\frac {1}{\sqrt {7}}},{\frac {1}{8}}<{\frac {1}{\sqrt {9}}}={\frac {1}{3}}}$ . More generally, for all ${\displaystyle n\in \mathbb {N} }$ there is ${\displaystyle 2n+1\leq 4n^{2}}$ (can be shown by induction) and hence ${\displaystyle a_{2n}={\frac {1}{2n}}={\frac {1}{\sqrt {4n^{2}}}}\leq {\frac {1}{\sqrt {2n+1}}}=a_{2n+1}}$. So the sequence of elements is not monotonously decreasing, since otherwise we would have ${\displaystyle a_{2n+1}\leq a_{2n}}$ . Only the subsequence of elements with even and with odd index would be monotonously decreasing on their own.
• However, ${\displaystyle (b_{k})}$ is a null sequence.

Exercise (Null sequence)

Prove it.

Proof (Null sequence)

The subsequences of elements with only odd or only even index are both null sequences : ${\displaystyle \lim _{n\to \infty }b_{2n-1}=\lim _{n\to \infty }{\frac {1}{\sqrt {2n-1}}}=0}$ and ${\displaystyle \lim _{n\to \infty }b_{2n}=\lim _{n\to \infty }{\frac {1}{2n}}=0}$. So

${\displaystyle \forall \epsilon >0\,\exists N_{1}\in \mathbb {N} \,\forall n\geq N_{1}:|b_{2n-1}-0|={\frac {1}{\sqrt {2n-1}}}<\epsilon \ }$ and ${\displaystyle \ \forall \epsilon >0\,\exists N_{2}\in \mathbb {N} \,\forall n\geq N_{2}:|b_{2n}-0|={\frac {1}{2n}}<\epsilon }$

(More precisely, some suitable ${\displaystyle N}$ numbers are provided by ${\displaystyle N_{1}>{\frac {1}{2\epsilon ^{2}}}+{\frac {1}{2}}}$ and ${\displaystyle N_{2}>{\frac {1}{2\epsilon }}}$.) Now, we set ${\displaystyle N=\max\{2N_{1}-1,2N_{2}\}}$. So if the index surpasses ${\displaystyle N}$ , it will also have surpassed ${\displaystyle 2N_{1}-1}$ and ${\displaystyle 2N_{2}}$, so both subsequences stay closer to 0 than ${\displaystyle \epsilon }$, which means that the entire sequence will stay closer to 0 than ${\displaystyle \epsilon }$:

${\displaystyle \forall \epsilon >0\,\exists N\in \mathbb {N} \,\forall k\geq N:|b_{k}-0|=b_{k}<\epsilon .}$

Therefore, ${\displaystyle (b_{k})}$ converges to ${\displaystyle 0}$.

Hence, the alternating series test does not apply. In fact, we can show that the sequence of partial sums ${\displaystyle (s_{2n})=\left(\sum _{k=1}^{2n}(-1)^{k}b_{k}\right)}$ is unbounded, so the series diverges. We use the estimate

${\displaystyle (k-1)^{2}\geq 0\Rightarrow k^{2}\geq 2k-1\Rightarrow k\geq {\sqrt {2k-1}}\Rightarrow {\frac {1}{\sqrt {2k-1}}}\geq {\frac {1}{k}}\Rightarrow {\frac {1}{\sqrt {2k-1}}}-{\frac {1}{2k}}\geq {\frac {1}{2k}}}$

This implies

{\displaystyle {\begin{aligned}s_{2n}&=\sum _{k=1}^{2n}(-1)^{k+1}b_{k}\\[0.5em]&=(b_{1}-b_{2})+(b_{3}-b_{4})+\ldots +(b_{2n-1}-b_{2n})\\[0.5em]&=\sum _{k=1}^{n}\left(b_{2k-1}-b_{2k}\right)\\[0.5em]&=\sum _{k=1}^{n}\left({\frac {1}{\sqrt {2k-1}}}-{\frac {1}{2k}}\right)\\[0.5em]&\left\downarrow \ {\text{estimate above}}\right.\\[0.5em]&\geq \sum _{k=1}^{n}{\frac {1}{2k}}\end{aligned}}}

Since the harmonic series diverges, also ${\displaystyle (s_{2n})}$ and the entire series will diverge. Loosely speaking, the reason for the divergence is that the series corresponding to the positive and negative elements, ${\displaystyle {\frac {1}{\sqrt {k}}}}$ and ${\displaystyle {\frac {1}{k}}}$ diverge to ${\displaystyle \infty }$ at different speeds. And their speed difference is so huge that the series of differences diverges, too.

Example (Example 3: A converging alternating series, which fails the alternating series test.)

Let us consider the series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ with ${\displaystyle b_{k}={\begin{cases}{\frac {1}{k^{2}}}&{\text{ for even }}k,\\0&{\text{ for odd }}k\end{cases}}}$. Now,

• ${\displaystyle (b_{k})}$ is not monotonously decreasing, as for all ${\displaystyle k\in \mathbb {N} }$ there is ${\displaystyle a_{2k-1}=0.

So the alternating series test does not apply. However, the series converges by direct comparison:

Exercise

Prove that ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges.

Solution

There is

• ${\displaystyle |(-1)^{k+1}b_{k}|\leq {\tfrac {1}{k^{2}}}}$ for all ${\displaystyle k\in \mathbb {N} }$
• ${\displaystyle \sum _{k=1}^{\infty }{\tfrac {1}{k^{2}}}<\infty }$

So the series converges by direct comparison, even though its elements do not form a monotonous sequence.

Conclusion: Error bounds for the limit

The alternating series test can show converges, but does not give us the limit. For instance, for the alternating harmonic series , there is ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}=\ln(2)}$ . But this limit can not be computed by the alternating series test. However, we can approximate the limit by considering partial sums and the alternating series test will provide us with a neat upper bound for the error of such an approximation.

We have seen above in this article, that the sequence of partial sums with odd index ${\displaystyle (S_{2n-1})}$ is monotonously decreasing and converges to the limit ${\displaystyle \lim _{n\to \infty }S_{2n-1}=S}$ . Further, ${\displaystyle S=\inf\{S_{2n-1}:n\in \mathbb {N} \}}$, where the infimum of a set is the greatest possible lower bound to its elements. Hence, ${\displaystyle S_{2n-1}\geq S}$ for all ${\displaystyle n\in \mathbb {N} }$, so we have upper bounds for the limit getting better and better. Conversely, ${\displaystyle (S_{2n})}$ is monotonously increasing with ${\displaystyle \lim _{n\to \infty }S_{2n}=S=\sup\{S_{2n}:n\in \mathbb {N} \}}$ . So ${\displaystyle S_{2n}\leq S}$ gives a lower bound for all ${\displaystyle n\in \mathbb {N} }$. That means, we have an estimate ${\displaystyle S_{2n}\leq S\leq S_{2n-1}}$ and ${\displaystyle S_{2n}\leq S\leq S_{2n+1}}$.

How good is the estimate? We subtract the two inequalities and get

{\displaystyle {\begin{aligned}S_{2n-1}-S&\leq S_{2n-1}-S_{2n}{\underset {\text{above}}{\overset {\text{see}}{=}}}b_{2n},{\text{ and}}\\S-S_{2n}&\leq S_{2n+1}-S_{2n}{\overset {\text{analogously}}{=}}b_{2n+1}.\end{aligned}}}

So, the series elements serve as a precision indicator for the estimate of the limit by partial sums:

${\displaystyle |S-S_{n}|=\left|\sum _{k=1}^{\infty }(-1)^{k+1}b_{k}-\sum _{k=1}^{n}(-1)^{k+1}b_{k}\right|\leq b_{n+1}.}$

Theorem (Error estimate for approximating alternating series)

If an alternating series ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k+1}b_{k}}$ converges by the alternating series test, then the limit can be approximated by the partial sums with maximum error

${\displaystyle \left|\sum _{k=1}^{\infty }(-1)^{k+1}b_{k}-\sum _{k=1}^{n}(-1)^{k+1}b_{k}\right|\leq b_{n+1}.}$

Example (Error estimate for approximating alternating series)

Let us try to get some numerical values for the limit of the alternating harmonic series ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}}$:

We take ${\displaystyle n=8}$ and get the estimate precision ${\displaystyle \left|\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}-\sum _{k=1}^{8}{\frac {(-1)^{k+1}}{k}}\right|\leq b_{9}={\frac {1}{9}}}$. Since ${\displaystyle S_{8}\leq S}$ , we know that the limit must be in the interval ${\displaystyle \left[{\frac {533}{840}},{\frac {533}{840}}+{\frac {1}{9}}\right]}$. We round this fractions to decimal numbers and obtain ${\displaystyle S\in [0.634,0.746]}$. In fact, the limit is ${\displaystyle S=\ln(2)\approx 0.693}$. However, this is only a coarse approximation which already needed summing up 8 terms. There are better ways to compute the logarithm, e.g. Newton's method. The good news is that the above estimate works for any series which converges by the alternating series test.

Generalizing the alternating series test to the Dirichlet test

The Dirichlet test serves for proving convergence of series of the form ${\displaystyle \sum _{k=1}^{\infty }a_{k}b_{k}}$ . It extends the alternating series test to cases where there is not ${\displaystyle a_{k}=(-1)^{k+1}}$ . This is particularly useful, if the presign does not change from element to element (like ${\displaystyle +,-,+,-,+,-,+,...}$) but can have streaks without a change in between (like ${\displaystyle +,-,-,+,-,+,+,...}$) . The proof is based on Abel's partial summation, which is quite some work to do. We will not state it here.

Theorem (Dirichlet's test)

Let ${\displaystyle (a_{k})}$ and ${\displaystyle (b_{k})}$ be real sequences with

• The partial sum sequence ${\displaystyle A_{n}=\sum _{k=1}^{n}a_{k}}$ being bounded and
• ${\displaystyle (b_{k})}$ being monotonously decreasing
• ${\displaystyle \lim _{k\to \infty }b_{k}=0}$.

Then, the series ${\displaystyle \sum _{k=1}^{\infty }a_{k}b_{k}}$ converges.

The conditions for ${\displaystyle (b_{k})}$ are exactly the same as for the alternating series test. Actually, with ${\displaystyle a_{k}=(-1)^{k+1}}$, we just get the alternating series test as a special case:

Exercise

Prove that ${\displaystyle a_{k}=(-1)^{k+1}}$ fulfils the first condition for Dirichlet's test, i.e.${\displaystyle A_{n}=\sum _{k=1}^{n}a_{k}}$ is bounded.

Solution

There is

${\displaystyle A_{n}=\sum _{k=1}^{n}a_{k}=\sum _{k=1}^{n}(-1)^{k+1}=1-1+1-1\pm \ldots +(-1)^{n}={\begin{cases}1&{\text{ for odd }}n,\\0&{\text{ for even }}n.\end{cases}}}$

So ${\displaystyle (A_{n})}$ alternates between 1 and 0 and is obviously bounded.