Skip to main content

Series and elementary function problems

Published

November 2, 2026

  1. Recall that we showed that \(\zeta(z)=\eta(z)/(1-2^{1-z})\). I’ve also stated (without proof) a couple of times that \[ \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \text{ and } \zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}. \] Use those facts to find closed-form expressions for \[ \eta(2) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \text{ and } \eta(4) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^4}. \]

  2. Use the definition of the exponential function to show that

    1. \(|e^z| \leq e^{|z|}\)
    2. \(\displaystyle \lim_{z\to0} \frac{e^z-1}{z} = 1\)

  3. Compute the values of the following:

    1. \(\text{Log}(\sqrt{3}+i)\)
    2. \(\log(8i)\)
    3. \((1+i)^{\pi i}\)
    4. \(2^i\)