If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
We have:
$\mathrm{S}_{\mathrm{p}}=1+\mathrm{r}^{\mathrm{p}}+\mathrm{r}^{2 \mathrm{p}}+\ldots \infty$
$\therefore \mathrm{S}_{\mathrm{p}}=\frac{1}{1-\mathrm{r}^{\mathrm{p}}}$
Similarly, $\mathrm{s}_{\mathrm{p}}=1-\mathrm{r}^{\mathrm{p}}+\mathrm{r}^{2 \mathrm{p}}-\ldots \infty$
$\therefore \mathrm{s}_{\mathrm{p}}=\frac{1}{1-\left(-\mathrm{r}^{\mathrm{p}}\right)}=\frac{1}{1+\mathrm{r}^{\mathrm{p}}}$
Now, $\mathrm{S}_{\mathrm{P}}+\mathrm{s}_{\mathrm{p}}=\frac{1}{1-\mathrm{r}^{\mathrm{P}}}+\frac{1}{1+\mathrm{r}^{\mathrm{p}}}=\frac{\left(1-\mathrm{r}^{\mathrm{p}}\right)+\left(1+\mathrm{r}^{\mathrm{p}}\right)}{\left(1-\mathrm{r}^{2 \mathrm{p}}\right)}$
$\Rightarrow \frac{2}{1-\mathrm{r}^{2 \mathrm{p}}}=2 \mathrm{~S}_{2 \mathrm{P}}$
$\therefore \mathrm{S}_{\mathrm{P}}+\mathrm{s}_{\mathrm{p}}=2 \mathrm{~S}_{2 \mathrm{P}}$
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