The sum to n terms of the series 1+21+1n+31+1n2+…...is given by

Let S be the sum of n terms of the given series and d x =1+1/n. Then, S=1+2x+3x2+4x3+…+nxn−1⇒xS=x+2x2+3x3+…+(n−1)xn−1+nxn∴S−xS=1+{x+x2+…+xn−1}−nxn⇒S(1−x)=1−xn1−x−nxn⇒S(−1/n)=−n[1−(1+1/n)n]−n(1+1/n)n⇒1nS=n{1−(1+1/n)n+(1+1/n)n}=n⇒S=n2