Series Convergence and Divergence II

In summary, the conversation discusses questions about different tests to determine the convergence or divergence of a series. For problem b, the series is compared to 1/(n^3/2) and found to be absolutely convergent by the p-test and comparison test. No further tests are needed. For problem a, the integral test is mentioned as another option, but it is shown that it does not work because the function is not always positive. This leads to the conclusion that the series is conditionally convergent. Finally, for problem c, it is established that the series is divergent by the ratio test.
  • #1
ardentmed
158
0
Hey guys,

I have a few more questions for the problem set I'm working on at the moment:
2014_07_15_320_5185576918465871c371_4.jpg

I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute divergence?

Also, for a, is the integral test another feasible test aside from the alternating series test? With the (-1) taken into account, the function would not be over positive values and thus fails the integral test. However, the absolute of the function would most definitely work. If so, the integral test concludes divergence, but the alternating series test is convergent. Thus, the series is conditionally convergent.

As for c, it's obviously divergent since (n+1)/e is infinity.
Thanks again.
 
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  • #2
ardentmed said:
I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute divergence?
No, you don't need other tests.

ardentmed said:
Also, for a, is the integral test another feasible test aside from the alternating series test? With the (-1) taken into account, the function would not be over positive values and thus fails the integral test. However, the absolute of the function would most definitely work. If so, the integral test concludes divergence, but the alternating series test is convergent. Thus, the series is conditionally convergent.
You are correct, the integral test establishes that the series is not absolutely convergent.

ardentmed said:
As for c, it's obviously divergent since (n+1)/e is infinity.
Yes, if by "obviously" you mean "by ratio test" and by "is" you mean "tends to".
 

FAQ: Series Convergence and Divergence II

What is the difference between a convergent and divergent series?

A convergent series is one whose terms eventually approach a finite limit, meaning that the sum of all the terms in the series is a finite number. In contrast, a divergent series is one whose terms do not approach a finite limit, meaning that the sum of the terms in the series either approaches infinity or does not have a defined value.

How can I determine if a series is convergent or divergent?

There are several tests that can be used to determine the convergence or divergence of a series, including the ratio test, the comparison test, and the integral test. These tests involve examining the behavior of the terms in the series and evaluating whether they approach a finite limit or not.

Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is convergent, it cannot also be divergent, and vice versa.

What is the importance of understanding series convergence and divergence in mathematics?

Series convergence and divergence are important concepts in mathematics because they allow us to determine the behavior of infinite sequences of numbers. This is particularly useful in applications such as calculus, where infinite series are used to represent functions and solve problems.

How can I use series convergence and divergence to solve real-world problems?

Series convergence and divergence can be applied to real-world problems in fields such as physics, economics, and engineering. For example, understanding the convergence or divergence of a series can help in predicting the behavior of a system over time or calculating the value of an investment over multiple periods.

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