Question

If S_p denotes the sum of the series 1 + r^p + r^2p + ... to ∞ and s_p the sum of the series 1 − r^p + r^2p − ... to ∞, prove that S_p + s_p = 2 . S_2p.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ {\mathrm{S}}_{\mathrm{p}}=1+{\mathrm{r}}^{\mathrm{p}}+{\mathrm{r}}^{2\mathrm{p}}+...\infty \\ \therefore {\mathrm{S}}_{\mathrm{p}}=\frac{1}{1-{\mathrm{r}}^{\mathrm{p}}} \\ \mathrm{Similarly},{\mathrm{s}}_{\mathrm{p}}=1-{\mathrm{r}}^{\mathrm{p}}+{\mathrm{r}}^{2\mathrm{p}}-...\infty \\ \therefore {\mathrm{s}}_{\mathrm{p}}=\frac{1}{1-\left(-{\mathrm{r}}^{\mathrm{p}}\right)}=\frac{1}{1+{\mathrm{r}}^{\mathrm{p}}} \\ \mathrm{Now},{\mathrm{S}}_{\mathrm{P}}+{\mathrm{s}}_{\mathrm{p}}=\frac{1}{1-{\mathrm{r}}^{\mathrm{p}}}+\frac{1}{1+{\mathrm{r}}^{\mathrm{p}}}=\frac{\left(1-{\mathrm{r}}^{\mathrm{p}}\right)+\left(1+{\mathrm{r}}^{\mathrm{p}}\right)}{\left(1-{\mathrm{r}}^{2\mathrm{p}}\right)} \\ \Rightarrow \frac{2}{1-{\mathrm{r}}^{2\mathrm{p}}}=2{\mathrm{S}}_{2\mathrm{P}} \\ \therefore {\mathrm{S}}_{\mathrm{P}}+{\mathrm{s}}_{\mathrm{p}}=2{\mathrm{S}}_{2\mathrm{P}} \end{gathered}$$