# need help proving $\sum_{i=1}^{\infty} (-1)^n\frac{e^{1/n}}{n^3}$ absolutely converges

We have the following infinite sum and must know whether it absolutely or conditionally converges or diverges

$$\sum_{i=1}^{\infty} (-1)^n\frac{e^{1/n}}{n^3}$$

I first proved it conditionally converges by the alternating series test, however my problem is proving if it absolutely converges.

I started off by taking the absolute value which gives

$$\sum_{i=1}^{\infty} \frac{e^{1/n}}{n^3}$$

I first tried the ratio test but the limit of the ratio of consecutive terms gives a value of 1, therefore inconclusive. I next tried the integral test, but as most of you may know $$e^{1/x}$$ has no elementary function which makes this test useless. Next i tried the limit comparison test with the general term $$\frac{e^n}{n^3}$$, but in order for that to work we need to prove that the general term we are comparing with converges, which i also had problems with(ratio test is inconclusive aswell and could not come up with a function to compare it with).

I am at a dead end and would like a hint towards proving this infinite series absolutely converges.

• $\frac{e^{\frac{1}{n}}}{n^3}$ is asymptotically equivalent to $\frac{1}{n^3}$, since $$\lim_{n \to \infty} \frac{e^{\frac{1}{n}}}{n^3} \cdot n^3=e^0=1$$ so $$\sum \frac{e^{\frac{1}{n}}}{n^3} < \infty$$ Apr 19 '19 at 9:16

$$|(-1)^{n}e^{1/n}| \leq e^{1}=e$$. Compare with $$\sum \frac 1 {n^{3}}$$.
• I see, so because the denominator dictates the behavior of the expression $\frac{e^{1/n}}{n^3}$ we must find an expression that also has a denominator that dictates the behavior of the expression very similarly regardless of the numerator in order to do the limit comparison test? Apr 19 '19 at 9:27