Abel limit theorem

I would like to know if the Abel limit theorem works if the limit is infinite. Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = \infty$ . Is it the case that $\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty$ ? Thank you in advance for any suggestions. I would just like to know if it is true, then I will try think of a proof myself.

Solution 1:

The idea I use here is a bit different from the answer given by Evan here because I don't need to show the radius of convergence of $\sum_ns_nx^n$ is $1$.

We assume that $\sum_{j=0}^{+\infty}a_j=+\infty$ and denote $s_n:=\sum_{j=0}^na_j$, with the convention $s_{-1}=0$. Then \begin{align} \sum_{j=0}^Na_jx^j&=\sum_{j=0}^N(s_j-s_{j-1})x^j\\ &=\sum_{j=0}^Ns_jx^j-\sum_{k=0}^{N-1}s_kx^{k+1}\\ &=s_Nx^N+(1-x)\sum_{k=0}^{N-1}s_kx^k. \end{align} Let $A$ be a positive number, arbitrary but fixed. Since $\sum_{j=0}^{+\infty}a_j=+\infty$, there is an integer $n_0$ such that $s_n\geqslant A$ whenever $n\geqslant n_0$. Hence, for $N\geqslant n_0$ and $0<x<1$, we get \begin{align} \sum_{j=0}^Na_jx^j&\geqslant s_Nx^N+(1-x)\sum_{k=0}^{n_0-1}s_kx^k+(1-x)\sum_{k=n_0}^NAx^k\\ &\geqslant Ax^N+(1-x)\sum_{k=0}^{n_0-1}s_kx^k+Ax^{n_0}(1-x^{N-n_0}). \end{align} Taking on both sides the limit $N\to +\infty$, this gives $$\sum_{j=0}^{+\infty}a_jx^j\geqslant (1-x)\sum_{k=0}^{n_0-1}s_kx^k+Ax_{n_0},$$ hence $$\liminf_{x\to 1^-}\sum_{j=0}^{+\infty}a_jx^j\geqslant A.$$ As $A$ was arbitrary, we reached the wanted conclusion.