Converge/Diverge: Show Work & Tests Used

[sdg612]

Determine whether the following converge or diverge (show all work and state all tests and theormes used!)

a)$$\infty$$ (n=1)$$\sum$$cos$$\hat{}$$2 n/n $$\sqrt{}$$n

b)$$\infty$$ (n=1)$$\sum$$4(-1)$$\hat{}$$n-1/n$$\hat{}$$5+n$$\hat{}$$3

i need help answering these 2 problems! any help wld be appreciated! Thnx!

 

[cristo]

Firstly, you need to show your work; we're not here to do your homework for you! Secondly, your expressions are not clear. Here, I'll rewrite them for you; click on the images to see the code:

$$\sum_{n=1}^{\infty}\frac{\cos^2n}{n\sqrt n}$$

$$\sum_{n=1}^{\infty}\frac{4(-1)^{n-1}}{n^5+n^3}$$

Are they the questions you are asking?

Finally, please be sure to post any further questions in the homework help forums.

Welcome to PF, BTW.

 

[sdg612]

thank you! yes those are the questions! it took me forever just to get them looking like that! Thank you again I'm still trying to get used to this forum...

 

[naqo]

Hi, i suposse i would procceed in the following way:
Note that due to the fact that cos(x) is always smaller or equal than 1,
cos^2(x) has the same behavior. So, the sum that you are asking:
$$\sum_{n=1}^{\infty}\frac{\cos^2n}{n\sqrt n}$$ is smaller than
$$\sum_{n=1}^{\infty}\frac{\1}{n\sqrt n}$$ and this last converges
because it's a "p-type" sum with p=3/2>1. So by comparission, the first sum
must converge.
In the second one, you may see that it's an alternant series (the 4 doesn't matter)
and also that the sequence formed by the terms is a positive and decreasing one, so
due to the Leibniz rule (that applies only in this case) the sum is convergent.

Sory for the english i still don't know it very well

 

[naqo]

there was supposed to be a 1 in the numerator of the second sum...