If Sp denote the sum of the series 1 - rp - r2p - - and Sp- the sum of the series 1 - rp - r2p - - prove that Sp - Sp- - 2S2p-

Given that S_p=1+r^p+r^2p+. . . . . . . . ∞ ............ (i) Replace p by 2p in above eqn, we get S_2p=1+r^2p+r^4p+. . .. . . . . . ∞ ...

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Byju's Answer

Standard XII

Mathematics

Sum of Infinite Terms of a GP

If Sp denot...

Question

If $S_{p}$ denote the sum of the series $1 + r^{p} + r^{2 p} + . . . . . + \infty$ and $S_{p}^{'}$ the sum of the series $1 - r^{p} + r^{2 p} - . . . . . . + \infty$ , prove that $S_{p} + S_{p}^{'} = 2 S_{2 p}$ .

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Solution

Given that
$S_{p} = 1 + r^{p} + r^{2 p} + . . . . . . . . \infty$ ............ $(i)$
Replace $p$ by $2 p$ in above eqn, we get
$S_{2 p} = 1 + r^{2 p} + r^{4 p} + . . . . . . . . . \infty$ .......... $(i i)$
$S_{p}^{'} = 1 - r^{p} + r^{2 p} - . . . . . . . \infty$ .......... $(i i i)$
Adding $(i)$ & $(i i i)$
$S_{p} + S_{p}^{'} = 2 + 2 r^{2 p} + 2 r^{4 p} + . . . . . . . . . . . . . . . . \infty$
$\Rightarrow S_{p} + S_{p}^{'} = 2 (1 + r^{2 p} + r^{4 p} + . . . . . . . . . . . . . . . . \infty)$
$\Rightarrow S_{p} + S_{p}^{'} = 2 (S_{2 p})$
Hence proved

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Similar questions

If $S_{p}$ denotes the sum of the series $1 + r^{p} + r^{2 p} + . . . .$ to $\infty a n d s_{p}$ the sum of the series $1 - r^{p} + r^{2 p} -$ ....to $\infty$ , prove that $S_{p} + s_{p} = 2 S_{2_{p}}$ .

Q. If

S_{p}

denote the sum of the series

1 + r^{p} + r^{2 p} + . . . \infty

and

S_{p}^{'}

denote the sum of the series

1 - r^{p} + r^{2 p} + . . . \infty

{S_{p}}^{'}

, then

S_{p} + {S_{p}}^{'}

equals

Q. 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.

Q. If

S_{p}

denotes the sum of the series

1 + r^{p} + r^{2 p} + . . . . . . .

\infty

and

s_{p}

denotes the sum of the series

1 - r^{p} + r^{2 p} - r^{3 p} + . . . . . . .

\infty, | r | < 1

then

S_{p} + s_{p}

Q. If

S_{p}

denote the sum of the series

1 + r^{p} + r^{2 p} + . .

upto infinity and

X_{p}

be the sum of the series

1 - r^{p} + r^{2 p} - . .

upto infinity then

(r \in (- 1, 1) - {0})

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If Sp denote the sum of the series 1+rp+r2p+.....+∞ and S′p the sum of the series 1−rp+r2p−......+∞, prove that Sp+S′p=2S2p.

If $S_{p}$ denote the sum of the series $1 + r^{p} + r^{2 p} + . . . . . + \infty$ and $S_{p}^{'}$ the sum of the series $1 - r^{p} + r^{2 p} - . . . . . . + \infty$ , prove that $S_{p} + S_{p}^{'} = 2 S_{2 p}$ .