Results 1 to 2 of 2

Math Help - Why does the series converge?

  1. #1
    Member Pranas's Avatar
    Joined
    Oct 2010
    From
    Europe. Lithuania.
    Posts
    81

    Why does the series converge?

    Hello.

    How to prove that the series converge?

    \[S = \sum\limits_{n = 2}^{ + \infty } {{{( - 1)}^n} \cdot {c_n}}  = \sum\limits_{n = 2}^{ + \infty } {\frac{{{{( - 1)}^n}}}{{\sqrt n  + {{( - 1)}^n}}}} \]

    That's obviously an alternating series because \[{c_n} > 0\]

    In addition \[\mathop {\lim }\limits_{n \to \infty } {c_n} = 0\]

    But \[{c_{n + 1}}\] is NOT always less than \[{c_n}\]

    and apparently it prevents me from using Leibniz's test.

    Without subtractions \[\sum\limits_{n = 2}^{ + \infty } {{c_n}} \] diverges so I've got no way to go

    Am I missing something obvious?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32
    You know that the series \sum_{n=2}^{+\infty}\frac{(-1)^n}{\sqrt n} is convergent. Hence S is convergent if and only if \sum_{n=2}^{+\infty}(-1)^nc_n-\frac{(-1)^n}{\sqrt n} is convergent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: June 26th 2010, 07:04 AM
  2. Series converge
    Posted in the Calculus Forum
    Replies: 7
    Last Post: November 27th 2009, 08:21 AM
  3. Does this series converge?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 5th 2008, 01:10 AM
  4. Does this series converge?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 3rd 2008, 09:31 PM
  5. When does the series converge
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 3rd 2008, 08:21 PM

/mathhelpforum @mathhelpforum