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 \(a\).)