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  4. Sum to infinity of the series 1+2 ⋅ (1/3)+3 ⋅ (1/32)+4 ⋅ (1/33)+ ldots . .

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Q. Sum to infinity of the series $1+2 \cdot \frac{1}{3}+3 \cdot \frac{1}{3^{2}}+4 \cdot \frac{1}{3^{3}}+\ldots . .$

KCETKCET 2022

Solution:

$S _{ x }=\frac{ a }{1- r }+\frac{ dr }{(1-r)^{2}}$
$=\frac{1}{1-(1 / 3)}+\frac{2(1 / 3)}{4 / 9}=\frac{9}{4}$