Telescoping sums and series – Serlo

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Telescoping series are certain series where summands cancel against each other. This makes evaluating them particularly easy.

Telescoping sums[Bearbeiten]

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 sum: Definition and explanation (YouTube video by the channel „MJ Education“)
Telescoping sums work like collapsing a telescope
A collapsible 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:

Example (telescoping sum)

Take the sum with and . We have

Or, putting a in front of everything:

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.

Telescoping series[Bearbeiten]

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 .

Theorem (convergence of telescoping series)

The telescoping series and converge if and only if the sequence converges. In that case, the limits are

and

Example (telescoping series)

The series diverges, since diverges.

However, the series converges, since the sequence converges to . The limit of the series is

Examples[Bearbeiten]

Example 1[Bearbeiten]

Exercise (Partial sums of the geometric series)

The aim of this exercise is to show the sum formula for geometric series without using induction. What we want to prove is for and . Show that the equivalent statement holds.

Solution (Partial sums of the geometric series)

For and there is

Example 2[Bearbeiten]

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

Example 3[Bearbeiten]

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.