Finding the Sum of Converging series when there is a range for $r$

Question

The problem is: what values of $x$ will cause the following series to converge:

$\sum_{n=0}^\infty 2^n (x+1)^n$

I wasn't sure how to determine x but just took a few moments of "playing" with the series to see what values would make for an $|r|\lt1$. I could tell I needed $-\frac 12\lt(x+1)\lt\frac 12$.

So, I would like to know the proper process for this rather than me just "playing with numbers".

Also, the next part of the problem said, find the sum of the series for those values of $x$. I'm really stuck on this because it appears they want ONE sum. Even if they want more than one sum, seems like the sum would have a range of values. I say this because with different values of $x$ I get various values of $r$, all of which are $\lt1$, making the series converge. But for $n=0$, the first term is always $1$, so it appears to me that I get a different sum for the series depending on what value of $x$ I use. And since $-1.5\lt x \lt -.5$, I can't use those values to evaluate the sum since it's greater than or less than not equal to.

Stuck here.

Answer 1

This is a simple geometric series with common ratio $r=2(1+x)$. So it is convergent iff $|2(1+x)| <1$ or $-\frac 3 2 < x <-\frac 1 2$ and the sum is $\frac 1 {1-r}=\frac 1 {1-2(1+x)} =-\frac 1 {1+2x}$.