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

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