The correct option is D n^2 Upon simplifying we get S=1+2(1 1n)^1+3(1 1n)^2...∞ Replacing (1 1n)=x and substituting in S, we get S=1+2x+3x^2 ...

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Roots of Quadratic Equation

∑k=1∞k1-1/nk-...

Question

$\infty \sum k = 1 k {(1 - \frac{1}{n})}^{k - 1} =$

n (n - 1)

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n (n + 1)

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n^{2}

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(n + 1)^{2}

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Solution

The correct option is D $n^{2}$
Upon simplifying we get
$S = 1 + 2 (1 - \frac{1}{n})^{1} + 3 (1 - \frac{1}{n})^{2} . . . \infty$
Replacing $(1 - \frac{1}{n}) = x$ and substituting in S, we get
$S = 1 + 2 x + 3 x^{2} . . . \infty$
$S = \frac{1}{(1 - x)^{2}}$
$= \frac{1}{(1 - 1 + \frac{1}{n})^{2}}$
$= \frac{1}{(\frac{1}{n})^{2}}$

$= n^{2}$

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