Question

If $S_p$ denotes the sum of the series $1+r^p+r^{2p}+....$ to $\infty and{s}_p$ the sum of the series $1-r^p+r^{2p}-$ ....to $\infty$, prove that $S_p+s_p=2S_{2_p}$.

Answer 1

$S_p=1+r^p+r^{2p}+...+\infty \therefore S_p=\frac{1}{1-r^p}{S}_p=1-r^p+r^{2p}+...+\infty \therefore S_p=\frac{1}{1+r^p}Now,S_p+s_p=\frac{1}{1-r^p}+\frac{1}{1+r^p}=\frac{(1-r^p)+(1+r^p)}{(1+r^{-2p})}=\frac{2}{1-r^{2p}}=2S_{2p}{S}_p+s_p=2\times S_{2p}=2S_{2p}$