Question

For what values of x will the infinite geometric series 1+ (2x-1) + (2x-1)^2 + (2x-1)^3 + ... have a finite sum?

Answer 1

This is a geometric series with common ratio `(2x-1)`.

In order to converge we require `-1 < (2x-1) < 1`

Hence: `0 < 2x < 2`

Hence: `0 < x < 1`

Note that if `x = 0` then partial sums will always be bounded, alternating between `1` and `0`, but the infinite series does not have a well defined sum.