Science:Math Exam Resources/Courses/MATH105/April 2015/Question 06 (b)/Hint 1

The radius of convergence of a power series (centred at 0) is the number $R$ for which the power series converges for all $|x|<R$ but diverges for all $|x|>R$.

The ratio test can be applied to determine the radius of convergence of such a power series: it says that the series converges when $\lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}x^{n+1}}{a_{n}x^{n}}}\right|<1$, and diverges when $\lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}x^{n+1}}{a_{n}x^{n}}}\right|>1$.