Improper Integral vs. Infinite Series

Math Online Tom Circle

$latex displaystyle int_{2}^{infty}{frac{1}{x^{2} – 1}dx} =boxed{ ln {3}}$

$latex displaystyle sum_{n=2}^{infty}{frac{1}{n^{2} – 1}} = boxed{frac{3 } {2 }} $

To test if the infinite series
$latex displaystyle sum_{n=2}^{infty}{frac{1}{n^{2} – 1}} $ converges, provided it is

  • positive &
  • decreasing function

you can convert it to $latex displaystyle int_{2}^{infty}{frac{1}{x^{2} – 1}dx} $
(although they converge to different values.)

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