Abs. conv, convergence, or divergence

In summary, the conversation discusses determining the convergence of a series, specifically the series \sum (-1)^n\frac{e^{1/n}}{n^4}. The speaker initially used the root test and concluded that the series diverges, but later realizes a mistake and uses the alternating series test to determine convergence. However, the speaker is unsure if the series is absolutely convergent and considers using the ratio or root test, but is reminded that other tests, such as the comparison test, can also be used.
  • #1
Puchinita5
183
0

Homework Statement



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

[tex]
\sum (-1)^n\frac{e^{1/n}}{n^4}
[/tex]

Homework Equations





The Attempt at a Solution


I used the root test so

[tex]
\sqrt[n]{\frac{e^{1/n}}{n^4}} -->
\lim_{n\to \infty }\frac{e}{n^{4/n}} = e> 1
[/tex]
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 ?
 
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  • #2
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.
 
Last edited:
  • #3
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.
 
  • #4
Puchinita5 said:
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.
 
  • #5
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.
 
  • #6
Puchinita5 said:
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.
 

FAQ: Abs. conv, convergence, or divergence

What is absolute convergence?

Absolute convergence is a term used in mathematics to describe a series that converges regardless of the order in which its terms are added. In other words, the sum of the series will always be finite, regardless of how the terms are rearranged.

What is conditional convergence?

Conditional convergence is a term used in mathematics to describe a series that only converges when its terms are added in a specific order. If the terms are rearranged, the sum of the series may be infinite.

How do you test for absolute convergence?

To test for absolute convergence, you can use the ratio test or the root test. These tests involve taking the limit of the absolute value of the terms in the series. If the limit is less than 1, the series is absolutely convergent. If the limit is greater than 1, the series is not absolutely convergent.

How do you test for conditional convergence?

To test for conditional convergence, you can use the alternating series test or the integral test. These tests involve checking if the series alternates between positive and negative terms, or if the series can be compared to an integral. If the series passes these tests, it is conditionally convergent.

What is the difference between absolute and conditional convergence?

The main difference between absolute and conditional convergence is the order in which the terms are added. Absolute convergence means the series converges regardless of the order of the terms, while conditional convergence means the series only converges if the terms are added in a specific order. Additionally, absolute convergence guarantees that the sum of the series is finite, while conditional convergence does not guarantee this.

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