test the series for convergence or divergence. \n∑(n = 1 to ∞) (e^n / n^2)\no convergent\no divergent\nneed…

Explanation:

Step1: Apply the ratio - test

Let (a_{n}=\frac{e^{n}}{n^{2}}). Then (a_{n + 1}=\frac{e^{n+1}}{(n + 1)^{2}}). Calculate the limit (L=\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_{n}}\right|). [ \begin{align*} \frac{a_{n + 1}}{a_{n}}&=\frac{\frac{e^{n+1}}{(n + 1)^{2}}}{\frac{e^{n}}{n^{2}}}\ &=\frac{e^{n+1}n^{2}}{e^{n}(n + 1)^{2}}\ &=e\cdot\left(\frac{n}{n + 1}\right)^{2} \end{align*} ]

Step2: Find the limit

We know that (\lim_{n\rightarrow\infty}\frac{n}{n + 1}=\lim_{n\rightarrow\infty}\frac{1}{1+\frac{1}{n}} = 1). Then (\lim_{n\rightarrow\infty}\left(\frac{n}{n + 1}\right)^{2}=1). So (L=\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_{n}}\right|=\lim_{n\rightarrow\infty}e\cdot\left(\frac{n}{n + 1}\right)^{2}=e> 1).

Answer:

divergent