Geometric series are series of the form
. They are important within several proofs in real analysis. In particular, they are crucial for proving convergence or divergence of other series. We will derive some criteria using them, e.g. the ratio or the root criterion.
Geometric sum formula[Bearbeiten]
We recall the geometric sum formula for partial sums of the geometric series. If you would like to know more about the geometric sum formula, take a look at the article „Geometrische Summenformel“ . The sum formula is proven there via induction. The proof of the sum formula reads as follows:
Proof (Geometrische Summenformel)
We have
The geometric series

converges for

,

or

.
We consider two cases:
and
.
Case
[Bearbeiten]
We consider the geometric series
for any
, which especially means
. The sum formula above applies to the partial sums in that case:
So the geometric series converges if and only if the sequence of partial sums
converges. This is the case if and only if
converges. We know that
converges to
if and only if
and it converges to
, if and only if
. In this section, we only care about the first case of convergence:
If
, then the geometric series
converges.
Now, let us determine its limit:
Alternatively, convergence for
can be shown directly, using the definition.
Exercise (alternative proof that the geometric series converges)
Prove that the geometric series
with
converges to
.
Case
[Bearbeiten]
For
, we have for all
, that
. Therefore, the sequence
cannot converge to 0. So teh series
must diverge (this argument is called term test and will be considered in detail, later)
The divergence becomes particularly obvious, if
is positive, e.g. for
.
In this case, for all
, we have
and may estimate the partial sums:
So the sequence of partial sums is bounded from below by the sequence
, which in turn diverges to
. So the series
must diverge, as well.
We have learned: for
,
and
, the geometric series diverges. These three cases can be concluded into one case
. However, if
, then the geometric series converges to
:
Solution (problems: geometric series)
Solution sub-exercise 1:
Solution sub-exercise 2:
Solution sub-exercise 3:
Solution (geometric series with special
)
Solution sub-exercise 1:
and
Solution sub-exercise 2:
and
Solution sub-exercise 3:
and
Solution (index shifting)
Solution sub-exercise 1:
Solution sub-exercise 2:
Solution (Sequences which relate to the geometric series)
Solution (Alternative proof to sub-exercise 3)
We may also add and subtract a 1:
Hint
Analogously to sub-exercise 3, one may show for every
that:
just replace
by