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

We will apply the ratio test to determine the radius of convergence of the power series $\sum _{n=0}^{\infty }a_{n}x^{n}$.

The ratio test says that the power series converges when

$\lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}x^{n+1}}{a_{n}x^{n}}}\right|=\lim _{n\rightarrow \infty }|x|\left|{\frac {a_{n+1}}{a_{n}}}\right|=|x|\lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|<1.$

By the solution to the Question 6(a), $\lim _{n\rightarrow \infty }a_{n}=1.$

Therefore, since each $a_{n}$ is positive,

$\lim _{n\rightarrow \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|=\lim _{n\rightarrow \infty }{\frac {a_{n+1}}{a_{n}}}={\frac {\lim _{n\rightarrow \infty }a_{n+1}}{\lim _{n\rightarrow \infty }a_{n}}}={\frac {1}{1}}=1.$

Hence, the series converges when $|x|<1.$ This shows that its radius of convergence is ${\color {blue}1}$.