Collection of Maths Problems

Series with parameters - absolute convergence

Task number: 2939

• Variant 1

  $\displaystyle \sum\limits_{n=1}^{\infty} \frac{a^n}{n}$

• Hint

  Use the ratio test for most cases.

• Resolution

  We first investigate absolute convergence using the ratio test.

  $\displaystyle \lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}= \lim_{n\to\infty}\frac{\left|\frac{a^{n+1}}{n+1}\right|}{\left|\frac{a^n}{n}\right|}= |a|\lim_{n\to\infty}\frac{|n|}{|n+1|}=|a|$.

  The series converges absolutely for $|a|<1$ and diverges for $|a|>1$.

  We must now investigate the case $|a|=1$. For $a=1$ we obtain the divergent harmonic series; for $a=-1$ we have the series $\displaystyle \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n}$, which converges conditionally by Leibniz's test.

• Result

  The series converges absolutely for $a\in (-1{,}1)$ and conditionally for $a=-1$; otherwise it diverges.

• Variant 2

  $\displaystyle \sum\limits_{n=1}^{\infty} \frac{a^n}{n\cdot3^n}$

• Result

  The series converges absolutely for $a\in (-3{,}3)$, conditionally for $a=-3$, otherwise diverges.

• Variant 3

  $\displaystyle \sum\limits_{n=1}^{\infty} \frac{n^2}{n^4+1}(a+1)^n$

• Result

  The series converges absolutely for $a\in \langle-2{,}0\rangle$, otherwise diverges.

• Variant 4

  $\displaystyle \sum\limits_{n=1}^{\infty} \frac{(a+2)^n}{\sqrt{n+1}}$

• Result

  The series converges absolutely for $a\in (-3{,}2)$, conditionally for $a=-3$, otherwise diverges.

• Variant 5

  $\displaystyle \sum\limits_{n=1}^{\infty} \frac{a^n\cdot n!}{n^n}$

• Hint

  Handle conditional convergence using Stirling's formula:

  $\displaystyle c_1{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}<{n!}<c_2{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}$, for certain constants $c_1$ and $c_2$.

• Result

  The series converges absolutely for $a\in (-e,e)$, otherwise diverges.

• Variant 6

  $\displaystyle \sum\limits_{n=1}^{\infty} n^{\ln a}$

• Result

  Converges absolutely for $a\in (0,\frac1e)$, diverges for $a\ge \frac1e$. ($\ln a$ is not defined for non-positive values of $a$.)