I conclude this series of posts by considering the formula for an infinite geometric series. Somewhat surprisingly (to students), the formula for an infinite geometric series is actually easier to remember than the formula for a finite geometric series.

One way of deriving the formula parallels the derivation for a finite geometric series. If are the first terms of an infinite geometric sequence, let

Recalling the formula for an geometric sequence, we know that

Substituting, we find

Once again, we multiply both sides by .

Next, we add the two equations. Notice that almost everything cancels on the right-hand side… except for the leading term . (Unlike yesterday’s post, there is no “last” term that remains since the series is infinite.) Therefore,

A quick pedagogical note: I find that this derivation “sells” best to students when I multiply by and add, as opposed to multiplying by and subtracting.

The above derivation is helpful for remembering the formula but glosses over an extremely important detail: not every infinite geometric series converges. For example, if and , then the infinite geometric series becomes

,

which clearly does not have a finite answer. We say that this series diverges. In other words, trying to evaluate this sum makes as much sense as trying to divide a number by zero: there is no answer.

That said, it can be shown that, as long as , then the above geometric series converges, so that

The formal proof requires the use of the formula for a finite geometric series:

We then take the limit as :

On the right-hand side, the only piece that contains an is the term . If , then as . (This limit fails, however, if or .) Therefore,

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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5 thoughts on “Formula for an infinite geometric series (Part 10)”

## 5 thoughts on “Formula for an infinite geometric series (Part 10)”