# Thread: How to prove thatseries converges (e.g., (n^2-1)/(n^4+1) from 1 to infinity)

1. ## How to prove a series converges (e.g., (n^2-1)/(n^4+1) from 1 to infinity)

Hi, I have a problem that is asking me to say whether several series converge or not, and to prove my claims. I am going to list all the series below... I would really appreciate it if you could solve one for me so that I can see how to do it; and in addition to this, if you could also give me a general explanation of what I have to look for or do to prove other series. Thanks a lot!

Series:

a) Sum from 1 to infinity of (n^2-1)/(n^4+1).
b) Sum from 1 to infinity of (n^(1/2))/(n^(3/2)+1).
c) Sum from 1 to infinity of 1/(n*(log n)^2).
d) Sum from 1 to infinity of n!/(n^n).

2. ## Re: How to prove a series converges (e.g., (n^2-1)/(n^4+1) from 1 to infinity)

Originally Posted by juanma101285
Series:

a) Sum from 1 to infinity of (n^2-1)/(n^4+1). limit comparison test with the known convergent series 1/n^2
b) Sum from 1 to infinity of (n^(1/2))/(n^(3/2)+1). limit comparison test with the known divergent series 1/n
c) Sum from 1 to infinity of 1/(n*(log n)^2). integral test
d) Sum from 1 to infinity of n!/(n^n). ratio test
try each recommended test for convergence