S(t)=\sum_{n=1}^{\infty}n\cdot t^{n+1}=\sum_{n=1}^{\infty}n\cdot t^{n-1+2}=\sum_{n=1}^{\infty}n\cdot t^{n-1}\cdot t^{2}=t^{2}\sum_{n=1}^{\infty}n\cdot t^{n-1}=t^{2}\sum_{n=1}^{\infty}(t^{n})^{'}=t^{2}(\sum_{n=1}^{\infty}t^{n})^{'}=t^{2}(\frac{t}{1-t})^{'}=(\frac{t}{1-t})^{2}