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Sum to $n$ terms of the series $S = 1 + 2 (1 + \frac{1}{n}) + 3 {(1 + \frac{1}{n})}^{2} + \dots$ is given by

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Correct option is A

Let x=1+1/n. Then S=1+2x+3x2…+nxn−1⇒ xS=x+2x2+…+(n−1)xn−1+nxnSubtracting, we get(1−x)S=1+x+x2+…+xn−1−nxn =1−xn1−x−nxn⇒ −1nS=(−n)1−1+1nn−n1+1nn=−n⇒ S=n2

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Sum to n terms of the series S=1+21+1n+31+1n2+⋯is given by

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Sum to $n$ terms of the series $S = 1 + 2 (1 + \frac{1}{n}) + 3 {(1 + \frac{1}{n})}^{2} + \dots$ is given by