Absolute-convergence of infinite series

Question

Precondition; limsup [n→∞] (|a_n|・|x-c|^n)^(1/n)=0 for all x.

Problem; Prove the fact that if the above precondition works, \sum_{n=0}^∞ |a_n・(x-c)^n|<∞ for all x.

I can't understand why \sum_{n=0}^∞ |a_n・(x-c)^n|<∞ (for all x) works.

Does anyone understand it and teach it to me ?

Answer 1

The hypothesis implies that $|a_n||x-c|^{n} <(\frac 1 2)^{n}$ for $n$ sufficiently large. Since $\sum (\frac 1 2)^{n}$ is convergent so is the given series.

Answer 2

Since $ \lim \sup (|a_n||x-c|^n)^{1/n}=0 <1$ for all $x$, the root test shows that $\sum_{n=0}^{\infty}a_n(x-c)^n$ converges absolutely for all $x$.