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agentx5

How would you evaluate this? \[\large \sum_{n=3}^{\infty} \frac{4}{n^2-1}\] o_O I'm thinking there's some trick here I'm not seeing...

  • one year ago
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  1. KingGeorge
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    My recommendation would be\[\sum_{n=3}^\infty \frac{4}{n^2-1}=\sum_{n=3}^\infty \frac{4}{(n+1)(n-1)}\]and then apply partial fractions.

    • one year ago
  2. agentx5
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    4 = A(n-1) + B(n+1) 4 = An - A + Bn + B A + B = 0 4 = -A + B A + (A+4) = 0 2A = -4 A = -2 B = 2 Now what?

    • one year ago
  3. agentx5
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    \[\sum_{n=3}^{\infty} \frac{-2}{n+1} + \sum_{n=3}^{\infty} \frac{2}{n-1}\]

    • one year ago
  4. KingGeorge
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    \[\sum_{n=3}^\infty \left(\frac{2}{n-1}-\frac{2}{n+1}\right)\]seems like a telescoping series to me.

    • one year ago
  5. agentx5
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    The kind where the middle terms keep cancelling each other out right? Yeah I was just introduced to that today :-D You're saying then all you have left are the first and last terms. Hmm, but how do I manipulate it into that form? I guess it the precise question here. Good call on the partial fractions btw, I didn't see that as an option at first glance

    • one year ago
  6. KingGeorge
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    It's already in that form. Just write out the first few terms \[\left(\frac{2}{3-1}-\frac{2}{3+1}\right)+\left(\frac{2}{4-1}-\frac{2}{4+1}\right)+\left(\frac{2}{5-1}-\frac{2}{5+1}\right)+...\]Simplify things a bit, and everything will cancel, and you'll be left with some value and nothing afterwards.

    • one year ago
  7. agentx5
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    Ah ok, that makes more sense because http://www.wolframalpha.com/input/?i=telescoping+series&lk=4 wasn't helping :P

    • one year ago
  8. KingGeorge
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    In this case, you don't even have to worry about anything at the end since \[\lim_{n\to\infty}\frac{4}{n^2-1}=0\]So the end term would be 0.

    • one year ago
  9. agentx5
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    +1000 Kudos! So if we can check if the limit goes to zero first thing off the bat then we're able to confirm convergence? Does that mean the evaluation of the series is also zero?

    • one year ago
  10. agentx5
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    Oh wait I think I may have misunderstood, the first few that don't cancel out are what will be left

    • one year ago
  11. agentx5
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    Yes?

    • one year ago
  12. KingGeorge
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    If the limit goes to 0 it doesn't guarantee convergence of the series. For example, \[\sum_{n=1}^\infty\frac{1}{n}\]does not converge. But the mere fact that it's telescoping with a finite number of non-zero terms means that it converges. And yes, the first few terms that don't cancel out will be the overall sum.

    • one year ago
  13. agentx5
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    Wonderful information, I'm evaluating it by hand at the moment. I'll post what I've found when I'm done. :-)

    • one year ago
  14. agentx5
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    I got \(\large\frac{5}{3}\) :-)

    • one year ago
  15. KingGeorge
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    That looks right.

    • one year ago
  16. agentx5
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    1 + 2/3 = 5/3

    • one year ago
  17. agentx5
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    Ty for helping out on this one :-D

    • one year ago
  18. KingGeorge
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    You're welcome.

    • one year ago
  19. klimenkov
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    I'm really sorry to interrupt you. But it is very interesting fact, that \[\sum_{n=3}^{\infty} \frac{4}{n^2-1}=\lim_{n \rightarrow \infty}\left( \frac 53 -\left( \frac{2}{n}+\frac{2}{n-1}\right) \right) \]I enjoy this!

    • one year ago
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