Abs. conv, convergence, or divergence

[Puchinita5]

Homework Statement

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

<br /> \sum (-1)^n\frac{e^{1/n}}{n^4}<br />

Homework Equations

The Attempt at a Solution

I used the root test so

<br /> \sqrt[n]{\frac{e^{1/n}}{n^4}} --&gt;<br /> \lim_{n\to \infty }\frac{e}{n^{4/n}} = e&gt; 1 <br />
so it should be divergent. Except my homework tells me I'm wrong. Where did I mess up? perhaps the limit as n approaches infinity of n^(4/n) is not 1 ?

 

[Dick]

n^(4/n) has limit 1 alright. But the nth root of e^(1/n) is (e^(1/n))^(1/n). That limit isn't e. The nth root test isn't going to tell you anything.

 

[Puchinita5]

i shoot. that was a silly mistake.

I just did an alternating series test and got that it was convergent. But how do i know it is "absolutely" convergent (which i know is the correct answer)? Wouldn't i need to use either the root or ratio test for that? And I tried the ratio test and got no where with that because I couldn't figure out a way to simplify.

hmmm.

 

[Dick]

i shoot. that was a silly mistake.

I just did an alternating series test and got that it was convergent. But how do i know it is "absolutely" convergent (which i know is the correct answer)? Wouldn't i need to use either the root or ratio test for that? And I tried the ratio test and got no where with that because I couldn't figure out a way to simplify.

hmmm.

You know the series 1/n^4 is absolutely convergent, right? It's a p-series. Try a comparison test.

 

[Puchinita5]

hmm i was under the impression that the only two things that could tell you absolute convergence were the ratio and root test. I knew the p-series was convergent for p>1 but i didn't think I could use that for absolute convergence. ::sigh:: I feel our professor really didn't teach these chapters very well.

 

[Dick]

hmm i was under the impression that the only two things that could tell you absolute convergence were the ratio and root test. I knew the p-series was convergent for p>1 but i didn't think I could use that for absolute convergence. ::sigh:: I feel our professor really didn't teach these chapters very well.

It's NOT true that the only thing you can use is the ratio test and the root test. 1/x^4 gives you an indeterminant result using both the ratio and the root test. Yet, you know it converges by an integral test. I still think this is an easy target for a comparison test. Try it. Don't let your professor hold you back.