Question

What is the interval of convergence of `sum [(-1)^n(2^n)(x^n)]`?

Answer 1

`(-1/2,1/2)`

Explanation:

First, note that `sum[(-1)^n2^nx^n] = sum(-2x)^n`

A geometric series of the form `sumr^n` converges if and only if `|r| < 1`. Then, for the given series to converge, we must have
`|-2x| < 1`, that is, `|x| < 1/2`. Thus the interval of convergence for the series is `(-1/2,1/2)`