Question

use any method to determine if the series converges or diverges. give reasons for your answer

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

oa. the series diverges because the limit used in the nth-term test is

ob. the series converges because the limit used in the root test is < 1.

oc. the series diverges by the comparison test if the series is compared with

od. the series diverges by the comparison test if the series is compared with

oe. the series converges because the limit used in the nth-term test is

of. the series converges because the limit used in the ratio test is < 1.

Explanation:

Step1: Apply the nth - Term Test

The nth - Term Test states that for a series \(\sum_{n = 1}^{\infty}a_{n}\), if \(\lim_{n
ightarrow\infty}a_{n}
eq0\), then the series \(\sum_{n = 1}^{\infty}a_{n}\) diverges.
For the series \(\sum_{n = 1}^{\infty}\frac{e^{n}}{n^{e}}\), we find \(\lim_{n
ightarrow\infty}\frac{e^{n}}{n^{e}}\).
We use L'Hopital's Rule. Consider the function \(y=\frac{e^{x}}{x^{e}}\), and take the natural logarithm: \(\ln y=x - e\ln x\).
Now, find \(\lim_{x
ightarrow\infty}(x - e\ln x)=\lim_{x
ightarrow\infty}x(1-\frac{e\ln x}{x})\).
We know that \(\lim_{x
ightarrow\infty}\frac{\ln x}{x}=\lim_{x
ightarrow\infty}\frac{1/x}{1}=0\) (by L'Hopital's Rule, since \(\lim_{x
ightarrow\infty}\frac{\ln x}{x}\) is in the \(\frac{\infty}{\infty}\) form). So, \(\lim_{x
ightarrow\infty}(x - e\ln x)=\infty\).
Since \(\lim_{x
ightarrow\infty}\ln y=\infty\), then \(\lim_{x
ightarrow\infty}y = \lim_{n
ightarrow\infty}\frac{e^{n}}{n^{e}}=\infty\).

Answer:

A. The series diverges because the limit used in the nth - Term Test is \(\infty\)