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

The Basic Divergence Detector rejects the series instantly, unless

$${\begin{aligned}0=\lim _{n\to \infty }[c-f(n)]=\lim _{n\to \infty }[c-1+(n+1)e^{-n}].\end{aligned}}$$

We can compute the following limit

$${\begin{aligned}\lim _{n\to \infty }(n+1)e^{-n}=\lim _{n\to \infty }{\frac {n+1}{e^{n}}},\end{aligned}}$$

by applying L'Hôpital’s Rule

$${\begin{aligned}\lim _{n\to \infty }{\frac {1}{e^{n}}}=0.\end{aligned}}$$

Then the only way for the following expression to hold true,

$${\begin{aligned}0=\lim _{n\to \infty }[c-1]\end{aligned}}$$

is if $c=1.$