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

This is a geometric series with common ratio $\left(2 x - 1\right)$.
In order to converge we require $- 1 < \left(2 x - 1\right) < 1$
Hence: $0 < 2 x < 2$
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.