Question

How do you find the sum of the infinite geometric series 2-2+2-2+..?

Answer 1

The series diverges by typical summation methods, but may converge depending on the method used.

Explanation:

In general, a geometric series will diverge if the common ratio `r` between terms does not have `|r| < 1`. In this case, `r = -1`, meaning `|r| = 1` and so the series should diverge.

Still, let's look at this case in a little more detail.

A series `sum_(k=0)^ooa_k` may be defined as `lim_(n->oo)sum_(k=0)^na_k`.

Typically, we refer to `sum_(k=0)^na_k` as the `n^(th)` partial sum of the series (often denoted `S_n`).

For the series in question, we have

`S_n = sum_(k=0)^n(-1)^k*2 = {(2" if n is even"),(0" if n is odd"):}`

As the sequence `(S_n)` simply alternates between `2` and `0`, it should be clear that `sum_(k=0)^oo(-1)^k*2= lim_(n->oo)S_n` does not converge to any value
(the fact that the sequence `(S_n)` does not converge may be shown simply using an `epsilon-delta` proof).

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The particular series mentioned is similar to a famous series known as Grandi's Series. Like Grandi's series, it diverges in conventional summation, but may converge if another method, such as Cesàro summation, is used.

Numberphile has a nice video on the subject here