1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence/Divergence of a Series

  1. Jul 11, 2008 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    3. The attempt at a solution
    I decided to use the ratio test:

    [tex]\frac{1}{2^{n+1}+(\frac{1}{3})^{n+1}}[/tex] x [tex]\frac{2^{n}+(\frac{1}{3})^{n}}{1}[/tex]

    And I got [tex]lim_{n\rightarrow\infty}[/tex][tex]\frac{1}{2 + \frac{1}{3}}[/tex]

    But I'm A) pretty sure it's wrong and B) if not, what do I do after that step?
  2. jcsd
  3. Jul 11, 2008 #2


    User Avatar
    Homework Helper

    You can't cancel the term [tex]2^n + (\frac{1}{3})^n[/tex] just like that since [tex]2^{n+1} + (\frac{1}{3})^{n+1} \ \mbox{is not} \ (2^n + (\frac{1}{3})^n) \cdot (2 + \frac{1}{3})[/tex]

    Instead express [tex]2^n + (\frac{1}{3})^n[/tex] as a fraction and do the same for the (n+1) expression as well. Something will cancel out and then you should be able to apply a certain limit rule to get the answer.
  4. Jul 11, 2008 #3
    I thought I could write 3n+1 as (3)(3n). So I should combine the two so that 2n + [tex]\frac{1}{3^{n}}[/tex] = [tex]\frac{7}{6}[/tex][tex]^{n}[/tex] and then find its limit?
  5. Jul 11, 2008 #4
    OH! Actually, could I write the f(x) as a fraction so I would get [tex]\frac{1}{\frac{7}{6}^{n}}[/tex] and then find its convergence/divergence through a geometric series? Or should I use an integral test?
  6. Jul 11, 2008 #5


    User Avatar
    Homework Helper

    3^(n+1) = 3(3^n). But that isn't what you're doing here. Just express [tex]2^{n+1} + \frac{1}{3^{n+1}}[/tex] as a single fraction. Then you can apply the ratio test easily.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Convergence/Divergence of a Series