Calculus Convergence/Divergence

Question

Which of the following series converge?

$\begin{array}{ll}\bf (i) & \sum\limits_{n=1}^{\infty}\dfrac{5^{n+1}}{6^n} \\[1.5ex] \bf (ii) & \sum\limits_{n=1}^{\infty} e^{1/n} \\[1.5ex] \bf (iii) & \sum\limits_{n=1}^{\infty}\dfrac{7^n}{5^{n+1}} \end{array}$

The answer I got was (iii) but it says I am wrong. Can anyone explain why it is wrong.

Answer 1

Trivially, both the second and third diverge by the basic divergence test.

If a series $\sum\limits_{n=1}^\infty a_n$ converges then $\lim\limits_{n\to\infty}a_n=0$

The contrapositive statement is:

If the terms of a series do not have zero as a limit then the series necessarily diverges.

In (ii) you have $\lim\limits_{n\to\infty} e^{\frac{1}{n}}=e^0=1\neq 0$

in (iii) you have $\lim\limits_{n\to\infty} \frac{7^n}{5^{n+1}}=\infty\neq 0$

The first actually does converge. It is what is known as a geometric series.

$\sum\limits_{n=1}^\infty \frac{5^{n+1}}{6^n}=5\sum\limits_{n=1}^\infty(\frac{5}{6})^n=25$