Show uniform convergent in $[-r,r]$

Question

I have to find the radius of convergence for $\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)x^n$, $x\in \mathbb R$.

I think it is: $r^{-1}$=$\lim_{n\to \infty}$$\frac{a_{n+1}}{a_n}$=1 $=>$ $r$=1. Then I have to show that $\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)x^n$ is uniform converget in the interval $[-r,r]$. I think I can use Weierstrass Majorant test. But I can't see what I can use as upper bound? Can anyone help?

It is the third question today about this series! Maybe next time it would be a great idea to go for an all-in-one question — Didier, May 28 '20 at 14:45
Does this answer your question? Radius of convergence and uniformly convergence — Noob mathematician, May 30 '20 at 21:06
You can accept answers to show that you appreciate the work of the people. — callculus42, Aug 05 '20 at 21:17

score 0 · Answer 1 · answered May 28 '20 at 14:35

0

Hint: show that if $0 \leq x \leq 1$, $0 \leq x-\sin{x} \leq x^3/6$.

answered May 28 '20 at 14:35

Aphelli

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Show uniform convergent in $[-r,r]$

1 Answers1