Tricki

## To calculate an infinite sum exactly, try antidifferencing

### Quick description

If you are trying to calculate the sum , then sometimes it is possible to spot another sequence such that and . If that is the case, then your sum is equal to .

### Prerequisites

The definition of an infinite sum.

 Contributions wanted This article could use additional contributions. Probably there is more to be said on this theme

### Example 1

An infinite sum that is well known to be straightforward to calculate exactly is the sum

The usual technique for summing this is to observe that , from which it follows that

which tends to 1 as tends to infinity.

This points to a circumstance in which an infinite sum can be evaluated exactly: we can work out the discrete analogue of an "antiderivative": that is, we have a sequence and can spot a nice sequence such that . Of course, must in that case be the partial sum (up to an additive constant). So is this really making the more or less tautologous observation that if you have a nice formula for the partial sums and can see easily what they converge to then you are done?

It isn't quite, because we could have spotted that without working out any partial sums.

### General discussion

Let us try to understand better why the above trick is not a universal method for calculating all infinite sums, by contrasting the above example with the example of the sum

Can we find some sequence such that ? It seems difficult just to spot such a sequence, so instead let us try to be systematic about it. It's not quite clear how to pick , so let us set it to be . Then , so . Next, , so . In general, we find that . Since we also want to tend to zero, this tells us what must be: , precisely the sum we were trying to calculate!

The reason the technique worked in the example above was that the partial sums turned out to have a nice formula.

Remark A simple method of generating undergraduate mathematics exercises is to reverse the telescoping-sums idea in order to create infinite sums that can be evaluated exactly. For example, take the sequence . This tends to as tends to infinity. Now let
The question you then ask is to calculate the sum . A smart student will use partial fractions to discover that this is a telescoping sum and will end up with the answer .

If this method works, it is a bit like managing to calculate an integral by antidifferentiating the integrand.