This question extends question Prove that pole of infinite series “explodes” to +∞ as I wonder if this proof technique still holds when the infinite series is defined as:
$$f(x) = \sum\limits_{k=1}^{\infty} \frac{kx^k}{(1-x^k)^2}, \quad x \in \mathbb{R}$$
How to proof that $f(-1) = -\infty$ ?
My attempts would be
$$\lim\limits_{t \to -1} \sum\limits_{k=1}^{\infty} \frac{k t^{k-1}}{(1-t^k)^2} = \sum\limits_{m=1}^{\infty} \left[ \lim\limits_{t \to -1} \frac{(2m-1) t^{2m-2}}{(1-t^{2m-1})^2} + \lim\limits_{t \to -1} \frac{2m t^{2m-1}}{(1-t^{2m})^2} \right] = \sum\limits_{m=1}^{\infty} \left[ \frac{(2m-1)}{4} - \infty \right] = -\infty$$
But I don't think I can do the proof in this way, because of $m \to \infty$.
EDIT 1: As DanielWainfleet said, my proof is not working in general. So, how can I proof it?