What is the radius of convergence R of the Taylor series?
The following is a taylor series of a function centered at 8.
$$\sum_{n=1}^{\infty}\frac{(-1)^n(x-8)^n}{7^n(n+8)}$$
I am trying to ind the radius of convergence of R of this series.
My guess is that the radius of convergence is $\infty$ because we have $(\frac{-1}{7})^n$, which approaches to 0 as n approaches $\infty$. Since we have a case of convergence, I suppose that the whole function should be convergent for whatever value of x. This is my attempt at answering the question, however, the answer seems to be incorrect.
$\endgroup$ 31 Answer
$\begingroup$Let $x_{n}$ be the sequence inside the sum. $\vert \frac{x_{n+1}}{x_{n}}|=\vert \frac{(x-8)(n+8)}{7(n+9)} \vert$. Hence, as n goes to infinity $\vert \frac{x_{n+1}}{x_{n}}|$ goes to $\vert \frac{(x-8)}{7} \vert$. The radius of convergence is 7.
$\endgroup$
