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

  1. #1
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    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).
    Last edited by juanma101285; October 16th 2011 at 02:02 PM.
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  2. #2
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    Re: How to prove a series converges (e.g., (n^2-1)/(n^4+1) from 1 to infinity)

    Quote Originally Posted by juanma101285 View Post
    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
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