Determine if $\sum_{n=1}^{\infty} \frac{1}{\exp(1/n)}$is convergent or divergent

Question

I have problem figuring out if this series is convergent or divergent. Please help!

$$\sum_{n=1}^{\infty} \frac{1}{\exp(1/n)}$$

Thanks

Answer 1

as n goes to infinity , $ e^{\frac{1}{n}} $ goes to 1. So your limit of summand goes to 1.. Therefore it should be divergent.

Answer 2

I'd use the Divergence Test to show that this infinite series diverges.

The theorem states that if a series ${a_n}$ does not converge to $0$, then $\sum_{n=1}^{\infty} a_{n}$ diverges.

$$\lim_{n\to\infty}\frac{1}{e^{\frac{1}{n}}} = \frac{1}{e^0} \to 1 \ne0 \implies a_{n} \ \ \text {diverges}$$