The-sum-to-infinity-of-the-series-1-3-1-15-1-35-is-1-2-

Question Number 56423 by gunawan last updated on 16/Mar/19

The sum to infinity of the series

\frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \dots . is \frac{1}{2} .

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19

T_{1} = \frac{1}{2} (\frac{3 - 1}{3 \times 1}) = \frac{1}{2} (1 - \frac{1}{3})

T_{2} = \frac{1}{2} (\frac{5 - 3}{5 \times 3}) = \frac{1}{2} (\frac{1}{3} - \frac{1}{5})

\dots

\dots

T_{n} = \frac{1}{2} {\frac{(2 n + 1) - (2 n - 1)}{(1 + (n - 1) \times 2) (3 + (n - 1) \times 2)}}

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

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

S_{\infty} = \frac{1}{2} (1 - 0) = \frac{1}{2}