Chapter 56: The Ratio Test

  • Let ρ=limnan+1an\rho = \lim_{n\rightarrow \infty} \frac{a_n +1}{a_n}
  • ρ<1n=1an|\rho| < 1 \Rightarrow \sum_{n=1}^{\infty}a_n  converges
  • ρ>1n=1an|\rho| > 1 \Rightarrow \sum_{n=1}^{\infty}a_n  diverges
  • ρ=1|\rho| = 1 means that the test is inconclusive


example #1

  • Determine if n=1(12)a \sum_{n=1}^{\infty}(\frac{1}{2})^a converges
  • Apply the ratio test


example #2

  • Show that n=11n! \sum_{n=1}^{\infty}\frac{1}{n!} converges

example #3

  • Determine whether n=1(2n)!n!n! \sum_{n=1}^{\infty}\frac{(2n)!}{n!n!} converges

Example #4

  • Discuss whether n=1n2(2n)! \sum_{n=1}^{\infty}\frac{n^2}{(2n)!} diverges
  • ρ=limn(n+1)2)(2n+2)!(2n)!n2\rho = \lim_{n \rightarrow \infty} \frac{(n+1)^2)}{(2n+2)!} \cdot \frac{(2n)!}{n^2}
  • =limn(n+1)2(2n+2)(2n+1)n2=0= \lim_{n \rightarrow \infty} \frac{(n+1)^2}{(2n+2)(2n+1)n^2} = 0


The Root Test

  • Let C=limn19n1nC = \lim_{n\rightarrow \infty} \sqrt[n]{19_n1}
  • If C<1C < 1  then n=1an\sum_{n=1}^{\infty}a_n converges
  • If C>1C > 1 then n=1an\sum_{n=1}^{\infty}a_n diverges
  • If C=1C = 1  then the test is inconclusive


Example #5

  • Determine whether n=1n22n\sum_{n=1}^{\infty}\frac{n^2}{2^n} converges
  • Apply the root test

  • Calculate limnn2n\lim_{n\rightarrow \infty} \sqrt[n]{n^2}

  • Calculate limn2nln(n)\lim_{n\rightarrow \infty} \frac{2}{n}ln(n)

example #6

  • Determine whether n=1(1n+1)1\sum_{n=1}^{\infty}(\frac{1}{n+1})^1 converges
  • Apply the root test

  • A series n=1an\sum_{n=1}^{\infty}a_n  converges absolutely if n=1an\sum_{n=1}^{\infty} |a_n|  converges


example #7

  • n=1(1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n} converges but does not converge absolutely


example #8

  • Check that n=1(1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n} converges and converges absolutely








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