If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$.
I found this question in Apostol. It can be compared to a result of Abel's theorem. What is the intuition behind this problem and how to solve this?
I saw this question in math stack exchange also:
If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$.
but I did not understand its solution.So I am looking for an intuition behind this problem.
Can someone provide me with a better solution for the same problem?
Hint: if $\sum_{n=1}^M a_n > N$, then $\sum_{n=1}^\infty a_n x^n \ge \sum_{n=1}^M a_n x^n > N$ if $x$ is close enough to $1$.