Real series/Quotient/Convergence/Quotient criteria/Exercise

Let ${}\sum _{n=0}^{n}a_{n}$ be a real series with ${}a_{n}\neq 0$ for all ${}n$. Suppose that the sequence of fractions

${}x_{n}={\frac {a_{n+1}}{a_{n}}}\,$

converge to a real number ${}x$ with ${}-1<x<1$. Show, using the ratio test, that the series converges.

Create a solution