Series comparison test question

In summary, the conversation involves determining which test to use for the summation from n=1 to infinity of (e^(1/n)) / n. The individual considers using a comparison test, with an = (e^(1/n))/n and bn = 1/n. They determine that since bn is a divergent p-series, an must also be divergent. They also note that an>bn. However, it is suggested that a better reason for e^(1/n)>1 should be given beyond just testing the first few terms.
  • #1
cue928
130
0

Homework Statement


Summation from n=1 to infinity: (e^(1/n)) / n


Homework Equations





The Attempt at a Solution



Can someone point out what criteria I should be considering when trying to determine which test to use? I was looking at a comparison test as a way to go on this one. I'd chosen an = (e^(1/n))/n and bn = 1/n. Since bn is a divergent p-series then an should also be divergent. Am I on the right track there?
 
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  • #2
Yes. an>bn, right?
 
Last edited:
  • #3
Yes, just testing out the first few terms, yes, an>bn, so the series an is divergent.
 
  • #4
cue928 said:
Yes, just testing out the first few terms, yes, an>bn, so the series an is divergent.

You might want to try to give a better reason why e^(1/n)>1 than just testing the first few terms.
 

FAQ: Series comparison test question

What is a series comparison test?

A series comparison test is a mathematical method used to determine if a given series is convergent (approaches a finite limit) or divergent (does not approach a finite limit). It involves comparing the given series to a known series with known convergence or divergence properties.

What are some examples of series comparison tests?

Examples of series comparison tests include the limit comparison test, ratio test, root test, and direct comparison test. These tests involve comparing the given series to a known series with a known convergence or divergence behavior.

How do I know which series comparison test to use?

Choosing the right series comparison test depends on the given series and its properties. Generally, the ratio test is used for series with factorials or exponentials, the root test for series with nth powers, and the direct comparison test for series with polynomials or rational expressions. However, it is important to understand the conditions and limitations of each test before applying them.

Can a series pass one comparison test and fail another?

Yes, a series may pass one comparison test and fail another. This is because each test has its own specific conditions and limitations. Therefore, it is important to use multiple tests to confirm the convergence or divergence of a series.

How can I use series comparison tests to prove the convergence of a series?

To prove the convergence of a series using comparison tests, you must show that the given series is less than or equal to a known convergent series. This can be done by manipulating the terms of the given series or by using algebraic techniques to rewrite the series in a more recognizable form. If the given series is less than or equal to a convergent series, then it must also be convergent.

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