JEE Main 2021 · Mathematics · Sequences and Series · medium

The sum of the infinite series 1/3+3/3^2+5/3^3+7/3^4+… is

Single correct option

The sum of the infinite series

\dfrac13+\dfrac33^2+\dfrac53^3+\dfrac73^4+\cdots

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Answer: (D) $1$

The series is an arithmetico-geometric one with general term

\dfrac{2n-1}{3^n}

. Let

S=\dfrac13+\dfrac{3}{3^2}+\dfrac{5}{3^3}+\cdots

. Then

\dfrac S3=\dfrac{1}{3^2}+\dfrac{3}{3^3}+\cdots

. Subtracting,

\frac{2S}{3}=\frac13+2\left(\frac1{3^2}+\frac1{3^3}+\cdots\right)=\frac13+2\cdot\frac{1/9}{1-1/3}=\frac13+\frac13=\frac23.

Hence

S=1

. As a check, the partial sums

0.333,\,0.556,\,0.704,\ldots

climb steadily towards

1

. The same value follows from

\sum nx^n=\dfrac{x}{(1-x)^2}

x=\dfrac13

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