The Problem: If $f(x)=\sum_{n=0}^{\infty}c_nx^n$ converges for $|x|<R$, then $f'(x)=\sum_{n=1}^{\infty}nc_nx^{n-1}$ in $(-R, R)$.
The problem is a theorem from Walter Rudin's PMA:
Here is part of Rudin's proof:
My Question: I don't quite follow the texts highlighted in yellow. The Root Test-which I suppose is what he is applying here-says that:
Thus, (3) converges implies only that $\underset{n\to\infty}{\lim}\sup\sqrt[n]{\vert c_nx^n\vert}\leq1$ by (b) of Theorem 3.33, from which we at most can deduce that $\underset{n\to\infty}{\lim}\sup\sqrt[n]{|n c_n x^{n-1}|}\leq 1$ (because we have $\underset{n\to\infty}{\lim}\sup\sqrt[n]{n\vert c_n\vert}=\underset{n\to\infty}{\lim}\sup\sqrt[n]{\vert c_n\vert}$, thus it seems $\underset{n\to\infty}{\lim}\sup\sqrt[n]{\vert c_nx^n\vert}=\underset{n\to\infty}{\lim}\sup\sqrt[n]{|n c_n x^{n-1}|}$). But what happens if $\underset{n\to\infty}{\lim}\sup\sqrt[n]{|n c_n x^{n-1}|}=1$? The same conundrum exists when (3) diverges.
Any help would be greatly appreciated.
Best Answer
You are looking at the root test in the wrong way. Instead of Theorem 3.3, you should take a look at Theorem 3.39 in Rudin, which tells you that the radius of convergence is determined by the limsup of the absolute value of the coefficients. Therefore, the identity $$ \limsup \sqrt[n]{n|c_n|}=\limsup \sqrt[n]{|c_n|} $$ tells you that the (3) converges if and only if (5) converges. Note that $R$ in the theorem you quote could be any number not bigger than the radius of convergence (the $R$ in Theorem 3.39).