Which of the following converge?

• \( \displaystyle \sum_{n=1}^\infty \frac{1}{\sqrt{n}} \)
• \( \displaystyle \sum_{n=1}^\infty \frac{3^n}{n!} \)
• \( \displaystyle \sum_{n=1}^\infty \left(\frac{e}{\pi}\right)^n \)

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The first series diverges since it is a \(p\)-series with \(p=\frac{1}{2}\leq1\)

The second series converges by the ratio test:

$$ \sum_{n=1}^\infty \frac{3^n}{n!} $$ $$ \lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} $$ $$ = \lim\limits_{n\to\infty} \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n} $$ $$ = \lim\limits_{n\to\infty} \frac{3}{n+1} $$ $$ = 0 $$

The third series is geometric with ratio \(\dfrac{e}{\pi} \lt 1 \)