Results 1 to 6 of 6

Math Help - Prove this series converges

  1. #1
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Prove this series converges

    Hi,

    I have the following question:

    Part d
    Using appropriate series convergence tests, prove that the series
    $\displaystyle\sum\limits_{n=1}^{\infty} \frac{\alpha^n}{\sqrt{n}}$
    converges if and only if the real number \alpha satisfies
    -1\le\alpha<1

    So my current working is this:

    Let a_n=\frac{\alpha^n}{\sqrt{n}}, then
    r= \frac{a_{n+1}}{a_n} =\frac{\alpha\sqrt{n}} {\sqrt{n+1}}

    Now as n tends to infinity, r will tend to \alpha.

    Therefore, by the ratio test:
    If \alpha<1, the series converges
    If \alpha>1, the series diverges
    If \alpha=1, a_n =\frac{1}{\sqrt{n}}
    In part b of this same question, we had to prove that \frac{1}{\sqrt{n}} diverges, which I did successfully.

    According to my working then, the series converges if \alpha<1, which only partly satisfies the requirement: -1\le\alpha<1

    Can someone spot what I've done wrong here?

    Thanks

    Note: In part c of the same question, we had to state and prove the Alternating Series Test, which I did successfully, so I'm thinking that we're meant to apply it here somehow, but I'm not sure how.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,391
    Thanks
    55

    Re: Prove this series converges

    Actually what you want using the ratio test is

     \lim_{n \to \infty} |\alpha \frac{\sqrt{n}}{\sqrt{n+1}}| < 1

    which gives | \alpha | < 1 or - 1 < \alpha < 1. Then check your endpoints giving the two series

    \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}

    and

    \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

    Then determine the convergence of each one to determine whether these endpoints should be included in your interval.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Re: Prove this series converges

    Quote Originally Posted by Danny View Post
    Actually what you want using the ratio test is

     \lim_{n \to \infty} |\alpha \frac{\sqrt{n}}{\sqrt{n+1}}| < 1

    which gives | \alpha | < 1 or - 1 < \alpha < 1. Then check your endpoints giving the two series

    \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}

    and

    \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

    Then determine the convergence of each one to determine whether these endpoints should be included in your interval.
    Thanks.
    According to Alternating series test - Wikipedia, the free encyclopedia, the Alternating Series Test is from n=0, not n=1. Does this make a difference?

    I previously knew the Alternating Series Test as
    \sum_{n=1}^{\infty}(-1)^{n+1}a_n
    Is this the same as what Wikipedia says?

    Thanks again
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,391
    Thanks
    55

    Re: Prove this series converges

    Quote Originally Posted by dwally89 View Post
    Thanks.
    According to Alternating series test - Wikipedia, the free encyclopedia, the Alternating Series Test is from n=0, not n=1. Does this make a difference?

    I previously knew the Alternating Series Test as
    \sum_{n=1}^{\infty}(-1)^{n+1}a_n
    Is this the same as what Wikipedia says?

    Thanks again
    You'll notice that \frac{1}{\sqrt{n}} is not defined at n = 0 so it should start at n = 1. As for the test, it doesn't matter shere you start. Adding or subtracting a few terms from the series won't change whether the series converges or not.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jul 2011
    Posts
    26

    Re: Prove this series converges

    Quote Originally Posted by Danny View Post
    You'll notice that \frac{1}{\sqrt{n}} is not defined at n = 0 so it should start at n = 1. As for the test, it doesn't matter shere you start. Adding or subtracting a few terms from the series won't change whether the series converges or not.
    Is "my version" of the alternating series test:
    \sum_{n=1}^{\infty}(-1)^{n+1}a_n
    wrong then?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,391
    Thanks
    55

    Re: Prove this series converges

    Nope - it doesn't matter.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How do i prove a series converges?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 2nd 2010, 07:35 PM
  2. Prove if series converges or diverges
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 3rd 2010, 06:29 PM
  3. prove that this series converges
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 14th 2009, 07:14 AM
  4. Replies: 2
    Last Post: September 10th 2009, 08:04 PM
  5. Prove that the series 1/n^2 converges
    Posted in the Calculus Forum
    Replies: 7
    Last Post: March 27th 2008, 08:42 PM

Search Tags


/mathhelpforum @mathhelpforum