Ratio Test

Section: 
9.4
Date: 
Wednesday, October 6, 2010 - 14:00 - 15:00
AttachmentSize
fall2010math2300_ratio-test.pdf42.3 KB

Math 2300 Section 005 – Calculus II – Fall 2010

9.4 Ratio Test Examples – October 6, 2010

  1. What does the ratio test say about the series?

    \sum _{{n=1}}^{\infty}\frac{1}{n}

    Solution:

    \lim _{{n\rightarrow\infty}}\frac{\displaystyle\frac{1}{n+1}}{\displaystyle\frac{1}{n}}=\lim _{{n\rightarrow\infty}}\frac{n}{n+1}=1

    and so the ratio test says nothing about the series. Of course, we already know that the harmonic series is divergent.

  2. What does the ratio test say about the series?

    \sum _{{n=1}}^{\infty}\frac{1}{n^{2}}

    Solution:

    \lim _{{n\rightarrow\infty}}\frac{\displaystyle\frac{1}{(n+1)^{2}}}{\displaystyle\frac{1}{n^{2}}}=\lim _{{n\rightarrow\infty}}\frac{n^{2}}{n^{2}+2n+1}=\lim _{{n\rightarrow\infty}}\frac{2n}{2n+2}=1

    and so again, the ratio test says nothing about the convergence of the series.

  3. What does the ratio test say about the series?

    \sum _{{n=1}}^{\infty}\frac{n}{2^{n}}

    Solution: Since

    \lim _{{n\rightarrow\infty}}\frac{\displaystyle\frac{n+1}{2^{{n+1}}}}{\displaystyle\frac{n}{2^{n}}}=\lim _{{n\rightarrow\infty}}\frac{2^{n}(n+1)}{n2^{{n+1}}}=\frac{1}{2}

    the ratio test says that the series converges.

  4. What does the ratio test say about the series?

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

    Solution:

    \displaystyle\lim _{{n\rightarrow\infty}}\frac{(n+1)!(n+2)!(2n)!}{(2(n+1))!n!(n+1)!}=\lim _{{n\rightarrow\infty}}\frac{(n+2)!(2n)!}{(2(n+1))!n!}
    \displaystyle\qquad\qquad=\lim _{{n\rightarrow\infty}}\frac{(n+2)(n+1)(2n)!}{(2n+2)!}
    \displaystyle\qquad\qquad=\lim _{{n\rightarrow\infty}}\frac{(n+2)(n+1)}{(2n+1)(2n+2)}
    \displaystyle\qquad\qquad=\lim _{{n\rightarrow\infty}}\frac{n^{2}+3n+2}{4n^{2}+6n+2}
    \displaystyle\qquad\qquad=\frac{1}{4}.

    Thus, the ratio test says in this case that the series converges.

© 2011 Jason B. Hill. All Rights Reserved.