Results 1 to 3 of 3

Math Help - Is this series convergent or divergent?

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    79

    Is this series convergent or divergent?

    Does this series converge or diverge?

    ( nthRoot(2) - 1 )

    I wanted to break it up into 2 series, but I don't think I can because I believe you must know that both parts are convergent to split it. Note that I am only allowed to use the integral test, test for divergence, comparison test, ratio test, alternating series test, and root test to find the answer, since we haven't learned anything else in class yet. If someone could just give me a hint as to what test to use or what to compare it to, that'd be great. Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Nov 2009
    Posts
    13
    In order to solve this problem, you need fo know that putting f(x)=2^x, f'(x)=(ln2)2^x.

    That being said, note that
    <br />
f'(0)=ln(2)=\lim_{x\to 0}\frac {f(x)-f(0)}{x},<br />
    hence
    <br />
ln(2)=\lim_{x\to 0}\frac {2^x-1}{x},<br />
    Putting x=1/n, you get
    <br />
ln(2)=\lim_{n\to +\infty}\frac {2^{1/n}-1}{1/n},<br />
    Since ln(2)>0,
    <br />
\left(\sum 1/n\right) and \left(\sum 2^{1/n}-1\right)<br />
    have the same nature.
    Since the first diverges (the harmonic series), your series diverges also.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    13
    I just realized you asked for a hint and not a complete solution. I'm sorry if I spoiled your exercise.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. series convergent or divergent?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 8th 2010, 11:35 AM
  2. Replies: 3
    Last Post: March 29th 2010, 12:36 PM
  3. Replies: 3
    Last Post: April 6th 2009, 11:03 PM
  4. convergent/divergent series
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 29th 2009, 08:08 PM
  5. Replies: 8
    Last Post: February 21st 2009, 10:16 AM

Search Tags


/mathhelpforum @mathhelpforum