Towards the end of the textbook Random Walk: A Modern Introduction by Gregory F. Lawler and Vlada Limic, the authors include a small section on Tauberian Theory, in which they state the following:
Suppose $u_n$ is a sequence of nonnegative real numbers. If $\alpha\in\mathbb{R}$, then the following two statements are equivalent: $$\sum_{n=0}^{\infty}\xi^nu_n=\bigg(\frac{1}{1-\xi}\bigg)\text{log}^\alpha\bigg(\frac{1}{1-\xi}\bigg),\xi\rightarrow1^{-}$$ $$\sum_{n=1}^{N}u_n\sim N\text{ }\text{log}^\alpha N, N\rightarrow\infty$$
They don't provide a proof of this though, and I can't find any online. How do I prove this statement, and what's the intuition behind it?