a-Find-the-sum-given-by-S-n-1-1-3-1-3-5-1-5-7-1-2n-1-2n-1-b-find-the-limit-of-S-n-as-n-

Question Number 8764 by tawakalitu last updated on 26/Oct/16

(a) Find the sum given by

S_{n} = \frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + \dots + \frac{1}{(2 n - 1) (2 n + 1)}

(b) find the limit of S_{n} as n \to \infty

Commented by sou1618 last updated on 26/Oct/16

S_{n} = \frac{1}{2} {(\frac{1}{1} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{5}) + \dots . + (\frac{1}{2 n - 1} - \frac{1}{2 n + 1})}

= \frac{1}{2} (1 - \frac{1}{2 n + 1})

= \frac{n}{2 n + 1}

{l i m}_{n \to \infty} \frac{n}{2 n + 1} = {l i m}_{n \to \infty} \frac{1}{(2 + \frac{1}{n})} = \frac{1}{2}

Commented by tawakalitu last updated on 26/Oct/16

Thank you sir . God bless you .