Results 1 to 2 of 2

Math Help - comparison test

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    30

    comparison test

    How would you work out these problems to find out if they converge or diverge?

    Problem 1

    n=1 to infinity 1/(square root (n^3 +2))

    Problem 2

    n=1 to infinity n+2^n/ n^2 2^n

    Problem 3

    n=1 to infinity 3^(n-1) + 1/ 3^n
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,912
    Thanks
    776
    Hello, TAG16!

    Converge or diverge?

    (1)\;\;S \;=\;\sum^{\infty}_{n=1} \frac{1}{\sqrt{n^3+2}}
    We have: . \frac{1}{\sqrt{n^3+2}} \:<\:\frac{1}{\sqrt{n^3}} \:=\:\frac{1}{n^{\frac{3}{2}}}


    \text{Hence: }\;\sum^{\infty}_{n=1}\frac{1}{\sqrt{n^3+2}} \quad <\; \!\!\! \underbrace{\sum^{\infty}_{n=1}\frac{1}{n^{\frac{3  }{2}}}}_{\text{convergent }p\text{-series}}

    Therefore, S converges.




    (2)\;\;S \;=\;\sum^{\infty}_{n=1} \frac{n+2^n}{n^2\!\cdot\!2^n}

    We have: . \sum \frac{n+2^n}{n^2\!\cdot\!2^n} \;=\;\sum\left(\frac{n}{n^2\!\cdot\!2^n} + \frac{2^n}{n^2\!\cdot\!2^n}\right) \;=\;\underbrace{\sum\frac{1}{n\!\cdot\!2^n}}_A + \underbrace{\sum\frac{1}{n^2}}_B

    Series A\!:\;\;\sum\frac{1}{n\!\cdot\!2^n} \:\leq\:\sum\frac{1}{2^n} . . . a convergent geometric series.
    . . Hence, A converges.

    Series B\!:\;\;\sum\frac{1}{n^2} . . . a convergent p-series.
    . . Hence, B converges.


    The sum of two convergent series is convergent.

    Therefore, S converges.




    (3)\;\;S \;=\;\sum^{\infty}_{n=1}\frac{3^{n-1} + 1}{3^n}

    We have: . \frac{3^{n-1}+1}{3^n} \;=\;\frac{3^{n-1}}{3^n} + \frac{1}{3^n} \;=\;\frac{1}{3} + \frac{1}{3^n} \;{\color{blue}\;>\;\frac{1}{3}}

    Hence: . \sum\frac{3^{n-1}+1}{3^n} \;> \;\sum\frac{1}{3} . . . a divergent series

    Therefore, S diverges.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Comparison test and Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 25th 2010, 01:54 AM
  2. Comparison or Limit Comparison Test Problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 12th 2010, 08:46 AM
  3. Limit comparison/comparison test series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 25th 2009, 09:27 PM
  4. Comparison & Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 25th 2009, 05:00 PM
  5. Integral Test/Comparison Test
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 27th 2009, 11:58 AM

Search Tags


/mathhelpforum @mathhelpforum