Question

does this sum converge: show steps

Answer 1

Answer:

Yes

Step-by-step explanation:

$\sum_{n=1}^{\infty} e^{-\sqrt{n} } \\\sum_{n=1}^{\infty} (\frac{1}{e})^{\sqrt{n} }$

If we say u = √n, then:

$\sum_{u=1}^{\infty} (\frac{1}{e})^{u}$

This is a geometric series.  Since |1/e| < 1, the series converges.