I'm trying to find the Power Series expansion and Radius of Convergence for the function $$f(z) = \frac{z}{e^{z} - 1}$$ at points $z = 0$ and $z = 1$.
When it comes to the radii of convergence, $e^{i\cdot(2n\pi)} = 1$ so the function has poles on the imaginary axis at intervals of $2\pi$. So, the series expansion centered at zero has radius of convergence equal to $2\pi$, whereas the series centered at $z=1$ has radius $1$ since the closest pole lies at $z = 0$.
Now, when it comes to actually finding the series expansion, I'm unsure how to find the $n^{th}$ derivative. How does one do so? And, is there a cleverer way of finding the series coefficients without computing the $n^{th}$ derivative?
But this only holds for $|P(z)| < 1$. I'm not sure how to show that this is true. Do you have any suggestions for doing this without integration? (My textbook gives this question before teaching integration).
– kdeoskar Dec 26 '23 at 15:46