Questions about series and sequences

[clickyclicky]

I have a test on this stuff and I'm confused about some things.

First, how do I show work to this problem?

I know that its absolute value diverge but I don't know how to show the work. Also not sure how to show the work for alternating series test. Not sure how to show that the it's decreasing.


I also have a question about sequences.

I should take the limit, but the sin throws me off.


Thanks,

clickyclicky

 

[Mark44]

I have a test on this stuff and I'm confused about some things.

First, how do I show work to this problem?

I know that its absolute value diverge but I don't know how to show the work.

Then how do you know this? If it's because the answer in the back of the book says so, then you are not very far along in working this problem.

Also not sure how to show the work for alternating series test.

What are the conditions that have to be satisfied so you can use the alternating series test?

Not sure how to show that the it's decreasing.

Let a_n = ln(n)/sqrt(n). Can you think of some calculus technique that would show whether this function of n is increasing or decreasing?

I also have a question about sequences.

How do I show that it diverges or converges? The sin throws me off.

3n - 1 <= 3n + sin(n) <= 3n + 1.

n <= n + sin²(n) <= n + 1.
Thanks,

clickyclicky

 

[rochfor1]

For the sequence, try multiplying the numerator and denominator by 1/n and then taking the limit. (Remember the Squeeze theorem.)

 

[clickyclicky]

For the first one I know that 1/√n diverges. If there's a ln n on the top it'll make the series slower.

If I try the alternating series test, I get 0 when I take the limit, but I can't figure out how to show that the function is decreasing. I tried to take the derivative but it doesn't tell me anything.


I'm not sure how to apply the squeeze theorem to the second one.

 

[rochfor1]

You don't evaluate the whole limit using the Squeeze theorem, just use it to show that sin n/n and sin^2 n/n go to 0.

 

[clickyclicky]

Are you saying that I should show that sin n is bounded and therefore goes to 0?

 

[rochfor1]

I'm saying exactly what I mean. I leave the details of the squeeze theorem to you.

 

[clickyclicky]

A quick look in the textbook and I see what you mean. Thanks.