Home » Blog » Find all z such that the series zn / n1/2 * log ((2n+1) / n) converges

Find all z such that the series zn / n1/2 * log ((2n+1) / n) converges

by RoRi

Find all complex numbers z such that the series

    \[ \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}} \log \frac{2n+1}{n} \]

converges.

First, we simplify the expression as follows,

    \begin{align*} \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}} \log \frac{2n+1}{n} &= \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}} \left( \log \left( 2 + \frac{1}{n} \right) \right) \\[9pt] &= \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}} \log 2 \left( 1 + \frac{1}{2n} \right) \\[9pt] &= \sum_{n=1}^{\infty} \left(\frac{z^n}{\sqrt{n}} \log 2 + \frac{z^n}{\sqrt{n}} \log \left( 1 + \frac{1}{2n} \right)\right). \end{align*}

Then, we consider the first series,

    \[ \log 2 \cdot \sum_{n=1}^{\infty}\frac{z^n}{\sqrt{n}}. \]

Let a_n = \frac{z^n}{\sqrt{n}} and use the ratio test,

    \begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{z^{n+1}}{\sqrt{n+1}} \right) \left( \frac{\sqrt{n}}{z^n} \right) \\[9pt] &= \lim_{n \to \infty} z \left( \frac{\sqrt{n}}{\sqrt{n+1}} \right) \\[9pt] &= z. \end{align*}

This series converges if |z| < 1 and diverges if |z| > 1. If |z|=1 and z \neq 1 then the series also converges by Dirichlet’s test. Finally, if z = 1, then the series is \sum \frac{1}{\sqrt{n}} which diverges.

Next, we know

    \[ \log \left( 1 + \frac{1}{2n} \right) < \log 2 < 1 \qquad \text{for all } n \geq 1. \]

Thus,

    \begin{align*} \sum_{n=1}^{\infty} \log \left( 1 + \frac{1}{2n} \right) \leq \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}}. \end{align*}

Hence, this series converges anywhere \sum_{n=1}^{\infty} \frac{z^n}{\sqrt{n}} converges. Therefore, the sum

    \[ \sum_{n=1}^{\infty} \left( \frac{z^n}{\sqrt{n}} \log 2 + \frac{z^n}{\sqrt{n}} \log \left( 1 + \frac{1}{2n} \right) \right) \]

converges for |z| \leq 1 with z \neq 1.

One comment

  1. S says:

    You should probably split complex a_n into the real and imaginary parts, prove they converge individually, and apply linearity to arrive to the final result. Ratio test requires positive sequence terms.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):