Telescoping series are certain series where summands cancel against each other. This makes evaluating them particularly easy.
Introductory example[Bearbeiten]
Consider the sum
Of course, we can compute all the brackets and then try to evaluate the limit when summing them up. However, there is a faster way: Some elements are identical with opposite pre-sign.
Every two terms cancel against each other. So if we shift the brackets (associative law), we get
This trick massively simplified evaluating the series. It works for any number
of summands:
This is called principle of telescoping sums: we make terms cancel against each other in a way that a long sum "collapses" to a short expression.
General introduction[Bearbeiten]
Telescoping sums work like collapsing a telescope
A telescoping sum is a sum of the form
. Neighbouring terms cancel, so one obtains:
analogously,
The name "telescoping sum" stems from collapsible telescopes, which can be pushed together from a long into a particularly short form.
Exercise
Prove that
.
Solution
There is
Definition and theorem[Bearbeiten]
Definition (telescoping sum)
A telescoping sum is a sum of the form
or
.
Theorem (value of a telescoping sum)
There is:
Partial fraction decomposition[Bearbeiten]
Unfortunately, most of the sums which can be "telescope-collapsed" do not directly have the above form, but must be brought into it. The following is an example:
The does not look like a telescoping sum: there is just one fraction. but there is a trick, which makes it a telescoping sum. For each
we have:
So
And this is a telescoping sum. Who would have guessed that ?!
The re-formulation
has a name: it is called partial fraction decomposition. A fraction with a product in the denominator is split into a sum, where each summand has only one factor in the denominator. This trick can serve in a lot of cases for turning a sum over fractions into a telescoping sum.
Introductory example[Bearbeiten]
What happens for infinitely many summands? Consider the series
The partial sums of this series are telescoping sums: For all
, there is:
So the limit amounts to
General introduction[Bearbeiten]
Telescoping series are series whose sequences of partial sums are telescoping sums. They have the form
. Their partial sums have the form
To see whether a telescoping series converges, we need to investigate whether the sequence
converges. This sequence in turn converges, if and only if
converges. If
is the limit of
, then the limit of the telescoping series amounts to
If
diverges, then
diverges, as well and the entire telescoping series diverges. Analogously, the series
converges, if we can show that
converges. In that case, the limit is
Definition, theorem and example[Bearbeiten]
Definition (telescoping series)
A telescoping series is a series of the form
or
.
Solution (Partial sums of the geometric series)
For
and
there is
Exercise
Does the series
converge? If yes, determine the limit.
Solution
We need a decomposition of the fraction here, if we want to make it a telescoping series. The denominator can be split in two factors, using the binomial theorems:
Now, we can do a partial fraction decomposition as above:
Hence, we get
Exercise
Does the series
converge? If yes, determine the limit.
Solution
Again there is only one fraction with a product in the denominator, so we attempt partial fraction decomposition:
This leads us to
Be careful: This series is not a telescoping series! We have to add summands - not to subtract them. Even worse, the series does not converge at all: The sequence of partial sums
is
So they are greater as for a diverging harmonic series
. By direct comparison,
diverges as well. So partial fraction decomposition does not necessarily produce a telescoping sum, but it can be useful to determine whether a series converges or diverges.
Series are sequences and vice versa[Bearbeiten]
In the beginning of the chapter, we have used that a series is actually nothing else than a sequence (of partial sums) Conversely, any sequence
can be made a series if we write it as a telescoping series: We can write
Question: Why is there
?
There is
So a sequence element can be written as

with
The sequence
can hence be interpreted as a series
, where the "series" is seen identical with "sequence of partial sums", here.