Question

The sum of series $2({1.7}^{-1}+3^{-1}{.7}^{-3}+5^{-1}{.7}^{-5}+......)$

Answer 1

$S=2({1.7}^{-1}+{3.7}^{-3}+{5.7}^{-5}+....)7^{-2}S=2(1{.7}^{-3}+{3.7}^{-5}+....)(1-7^{-2})S=2({1.7}^{-1}+{2.7}^{-3}+{2.7}^{-5}+....)\frac{485}{49}S=\frac{2}{7}+\frac{4}{7^3}(1+\frac{1}{7^2}+....)\frac{485}{49}S=\frac{2}{7}+\frac{4}{7^3}\left(\frac{49}{48}\right)\frac{485}{49}S=\frac{2}{7}+\frac{4}{7\left(48\right)}\frac{485}{49}S=\frac{25}{84}S=\left(\frac{35}{97\times 12}\right)$