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  4. If the sum to infinity of the series 1+2 r+3 r2+4 r3+ ldots is 9 / 4, then value of r is

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Q. If the sum to infinity of the series $1+2 r+3 r^{2}+4 r^{3}+\ldots$ is $9 / 4$, then value of $r$ is

Sequences and Series

Solution:

$S=1+2 r+3 r^{2}+4 r^{3}+\ldots$
$r S=r+2 r^{2}+3 r^{3}+4 r^{4}+\ldots$
Subtracting,
$(1-r) S=1+r+r^{2}+r^{3}+\ldots$
$=\frac{1}{1-r}$
$\Rightarrow S =\frac{1}{(1-r)^{2}}$
Given $S =9 / 4 \Rightarrow \frac{1}{(1-r)^{2}}=9 / 4$
$\Rightarrow 1-r=\pm \frac{2}{3}$
$\Rightarrow r=1 / 3$ or $5 / 3$
Hence $r =1 / 3$ as $-1 < r < 1$