Let $\sum_{1}^∞ a_n*x^n$ be a power series with radius of convergence 2 and note that the constant term is 0. Show that there is a constant P so that |$\sum_{1}^∞ a_n*x^n$|< $Px$ for every x satisfying $|x| ≤ 1$.
Couldn't find anything in my notes to help with this. I get that it has to be less than something, given that it converges but not sure how to prove that it is less than $Px$
Use $\vert\sum_{1}^∞ a_n*x^n\vert\leq \sum_{1}^∞ \vert a_n\vert*\vert x\vert^n$ and $\vert x\vert^n\leq \vert x\vert$ for $\vert x\vert\leq 1$.
The sum of the series (call it $f$) is of class $C^\infty$ in any interval $(-a,a)\subset[-2,2]$. Therefore, $|f'|$ is bounded on $(-1,1)$ by some $P\ge 0$. By the mean value theorem on $[0,x]$ with $0\le x\le 1$ (or $[x,0]$ if $-1\le x\le 0$) you have $$ |f(x)-f(0)|=|f(x)|=|f'(c)||x|\le P|x| $$ for any $x\in[-1,1]$.
