Abel convergence test

[takelight]

So I am trying to prove the convergence of the following series using the Abel test:

- The test postulates as follows:
Abel’s Test:
Let {an} and {bn} be two sequences. Suppose the following statements are true:
- Sum(∞) n=1 {an} is convergent.
- {bn} is bounded and monotonic.
Then Sum(∞) n=1 {an}{bn} is convergent.

Prove the convergence of the following:

1.





For this one, I chose [((-1)^n)/sqrt(n)] for {an} and as {bn}. I found {an} to be conditionally convergent, but I am having trouble proving that cos(1/n) is bounded and monotonic. I don't think it is... Help pls!!


2.



For this one, I chose an to be the first product, and bn to be the second sum. The problem is that the definition says that bn has to be a sequence, how do I go about showing the convergence of this one??

 

[pka]

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[takelight]

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Should be visible now. Thanks

 

[pka]

So I am trying to prove the convergence of the following series using the Abel test:

- The test postulates as follows:
Abel’s Test:
Let {an} and {bn} be two sequences. Suppose the following statements are true:
- Sum(∞) n=1 {an} is convergent.
- {bn} is bounded and monotonic.
Then Sum(∞) n=1 {an}{bn} is convergent.

Prove the convergence of the following:

1.


View attachment 17603


For this one, I chose [((-1)^n)/sqrt(n)] for {an} and as {bn}. I found {an} to be conditionally convergent, but I am having trouble proving that cos(1/n) is bounded and monotonic. I don't think it is... Help pls!!


2.View attachment 17604



For this one, I chose an to be the first product, and bn to be the second sum. The problem is that the definition says that bn has to be a sequence, how do I go about showing the convergence of this one??

For #1 note that

So I am trying to prove the convergence of the following series using the Abel test:

- The test postulates as follows:
Abel’s Test:
Let {an} and {bn} be two sequences. Suppose the following statements are true:
- Sum(∞) n=1 {an} is convergent.
- {bn} is bounded and monotonic.
Then Sum(∞) n=1 {an}{bn} is convergent.

Prove the convergence of the following:

1.


View attachment 17603


For this one, I chose [((-1)^n)/sqrt(n)] for {an} and as {bn}. I found {an} to be conditionally convergent, but I am having trouble proving that cos(1/n) is bounded and monotonic. I don't think it is... Help pls!!


2.View attachment 17604



For this one, I chose an to be the first product, and bn to be the second sum. The problem is that the definition says that bn has to be a sequence, how do I go about showing the convergence of this one??

Note that \(\mathop {\lim }\limits_{x \to {0^ + }} \cos (x) = 1\). look at the graph of the \(\cos(x)\) Note the derivative is \(-\sin(x)\) meaning what?

 

[takelight]

For #1 note that

Note that \(\mathop {\lim }\limits_{x \to {0^ + }} \cos (x) = 1\). look at the graph of the \(\cos(x)\) Note the derivative is \(-\sin(x)\) meaning what?

I see, so since the limit reaches 1, the sequence cos(1/n) would be getting closer to 0. So, it is bounded. That's what I missed. As for being monotonic, the sequence is decreasing, and so it is also monotonic. Thanks for the help. One thing though... for the Abel test to be valid, does the first sequence have to be absolutely convergent?

 

[pka]

I see, so since the limit reaches 1, the sequence cos(1/n) would be getting closer to 0. So, it is bounded. That's what I missed. As for being monotonic, the sequence is decreasing, and so it is also monotonic. Thanks for the help. One thing though... for the Abel test to be valid, does the first sequence have to be absolutely convergent?

Actually \(\cos\left(\frac{1}{n}\right)\) is increasing as \(n\to\infty\).
It derivative is negative, think about it. [the points are "backing-up"] LOOK HERE