Determine whether the following SERIES converge or diverge

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In summary, for the first series, the ratio test was used and it was found that the limit is less than 1, indicating that the series converges absolutely. For the second series, the limit comparison test was used, and it was determined that it diverges. The comparison series was not explicitly stated, but it can be assumed that it was used to show that the given series also diverges.
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shamieh
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can someone check my answers

Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

4.\(\displaystyle \sum^{\infty}_{n=1} \frac{n^3}{3^n}\)

A: So for this one i used the ratio test and found that L = 1/3 which is < 1 therefore the series converges absolutely

5. \(\displaystyle \sum^{\infty}_{n = 1} \frac{n}{n^2 + 1}\)

A: diverges by limit comparison test
 
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shamieh said:
4.\(\displaystyle \sum^{\infty}_{n=1} \frac{n^3}{3^n}\)

A: So for this one i used the ratio test and found that L = 1/3 which is < 1 therefore the series converges absolutely
Correct. (There is no need to add "absolutely" here since all terms are positive.)

shamieh said:
5. \(\displaystyle \sum^{\infty}_{n = 1} \frac{n}{n^2 + 1}\)

A: diverges by limit comparison test
The problem statement asked to explain your answer. Mentioning the limit comparison test but not saying with what series you compare the given one is not a sufficient explanation. It would be nice to also say how you know that the second series converges or diverges.
 

FAQ: Determine whether the following SERIES converge or diverge

What is a series?

A series is a sum of the terms of a sequence. It can be finite or infinite.

What does it mean for a series to converge or diverge?

A convergent series is one where the sum of its terms approaches a finite value as the number of terms increases. A divergent series is one where the sum of its terms does not approach a finite value and may increase without bound.

How do you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, such as the ratio test, the comparison test, and the integral test. These tests involve evaluating the behavior of the series as the number of terms increases.

What is the importance of determining if a series converges or diverges?

Determining if a series converges or diverges is important in understanding the behavior of the terms in the series. It allows us to determine the overall sum of the series and whether it can be used to approximate a value. It also has applications in various fields such as physics, engineering, and economics.

Can a series both converge and diverge?

No, a series can only either converge or diverge. It cannot do both simultaneously.

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