radius of convergence involving power series

Question

Consider the power series $$\sum_{n=1}^{\infty}\left[x^n(1-\cos\left(1 \over n\right))\right].$$

(I) Determine its radius of convergence $R$

(II) Examine the convergence/divergence of the power series when $x = +R$ and $x = -R$

Help will be much appreciated. Im not sure how to proceed

Answer 1

If we use the well known equivalence

$$1-\cos(X)\sim \frac{X^2}{2}\;\;\;(X\to 0)$$

with $X=\frac 1n$ and $n\to +\infty$,

we get

$$1-\cos(\frac 1 n)\sim \frac{1}{2n^2}$$

the two series have the same radius $R=1,$ by ratio test

at $x=\pm1, $

$\sum \frac{1}{n^2}$ and $\sum (1-\cos(\frac{1}{n})) (-1)^n $ converge absolutly by the limit comparison test.