Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 20 (c)/Solution 1

In part b) we found the constant to be $c=1$. Thus, we seek the exact value of $S=\sum _{n=0}^{\infty }(n+1)e^{n}.$ This is related to the function $${\begin{aligned}f(x)=\sum _{n=0}^{\infty }(n+1)x^{n},\end{aligned}}$$ at the value at $x=e^{-1}.$

To evaluate it, we use known power series results. We recall the sum of a geometric series: $${\begin{aligned}\sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}},\quad {\rm {for}}\;|x|<1.\end{aligned}}$$

To generate the term $(n+1)x^{n}$, differentiate both sides of the geometric series with respect to $x$

$${\begin{aligned}f(x)=\sum _{n=0}^{\infty }(n+1)x^{n}={\frac {d}{dx}}{\frac {1}{1-x}}={\frac {1}{(1-x)^{2}}},\quad x\in (-1,1).\end{aligned}}$$ The point $x=e^{-1}$ lies in the interval of convergence, so we can substitute it

$${\begin{aligned}S=f(e^{-1})={\frac {1}{(1-e^{-1})^{2}}}={\frac {e^{2}}{(e-1)^{2}}},\end{aligned}}$$ obtaining the exact value of $S.$