Convergence and Divergence Tests for Series: Real Analysis Homework

[rayred]

Homework Statement

a) Show that the series ∑ from n = 1 to infinity 1/n^p where p converges when p > 1 and
diverges for p=1.

b) Prove that the following series diverges: ∑ from n = 1 to infinity sqrt(n)/n+1

c) Use an appropriate test to show whether ∑ from n = 1 to infinity [(−1)^n * n^2/(n^2 +1)] converges or diverges.

d) For what values of x , if any, does the following series converge? Show
how you arrived at your answer.
∑ from n = 1 to infinity [(x^(2n + 1))/(2n + 1)!]

The Attempt at a Solution

Im lost completely

 

[LCKurtz]

Homework Statement

a) Show that the series ∑ from n = 1 to infinity 1/n^p where p converges when p > 1 and
diverges for p=1.

b) Prove that the following series diverges: ∑ from n = 1 to infinity sqrt(n)/n+1

c) Use an appropriate test to show whether ∑ from n = 1 to infinity [(−1)^n * n^2/(n^2 +1)] converges or diverges.

d) For what values of x , if any, does the following series converge? Show
how you arrived at your answer.
∑ from n = 1 to infinity [(x^(2n + 1))/(2n + 1)!]

The Attempt at a Solution

Im lost completely

I'm sorry to hear that. If you are that lost now at the end of the semester, I'm guessing you will be repeating your course. You aren't going to find anyone here to work those for you without you showing some work of your own.

 

[sid9221]

a)
http://dl.dropbox.com/u/33103477/222.jpg
http://dl.dropbox.com/u/33103477/222222.jpg

b)Use a comparison test, (n+1)>(n+1)/sqrtn so 1/(n+1)<sqrt(n)/(n+1).
Now 1/(n+1) diverges so the series you want also diverges.

c) Use the fact that an absolutely convergent series converges, so just prove, n^2/n^2+1 convereges to 1 using limits hence the whole damn thing converges.

d)try a ratio test

 

[rayred]

@sid Thank you a bunch!