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 2 years ago
How would you evaluate this?
\[\large \sum_{n=3}^{\infty} \frac{4}{n^21}\]
o_O I'm thinking there's some trick here I'm not seeing...
 2 years ago
How would you evaluate this? \[\large \sum_{n=3}^{\infty} \frac{4}{n^21}\] o_O I'm thinking there's some trick here I'm not seeing...

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KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.3My recommendation would be\[\sum_{n=3}^\infty \frac{4}{n^21}=\sum_{n=3}^\infty \frac{4}{(n+1)(n1)}\]and then apply partial fractions.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.14 = A(n1) + 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?

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1\[\sum_{n=3}^{\infty} \frac{2}{n+1} + \sum_{n=3}^{\infty} \frac{2}{n1}\]

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.3\[\sum_{n=3}^\infty \left(\frac{2}{n1}\frac{2}{n+1}\right)\]seems like a telescoping series to me.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1The 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

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.3It's already in that form. Just write out the first few terms \[\left(\frac{2}{31}\frac{2}{3+1}\right)+\left(\frac{2}{41}\frac{2}{4+1}\right)+\left(\frac{2}{51}\frac{2}{5+1}\right)+...\]Simplify things a bit, and everything will cancel, and you'll be left with some value and nothing afterwards.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1Ah ok, that makes more sense because http://www.wolframalpha.com/input/?i=telescoping+series&lk=4 wasn't helping :P

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.3In this case, you don't even have to worry about anything at the end since \[\lim_{n\to\infty}\frac{4}{n^21}=0\]So the end term would be 0.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1+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?

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1Oh wait I think I may have misunderstood, the first few that don't cancel out are what will be left

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.3If 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 nonzero terms means that it converges. And yes, the first few terms that don't cancel out will be the overall sum.

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1Wonderful information, I'm evaluating it by hand at the moment. I'll post what I've found when I'm done. :)

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1I got \(\large\frac{5}{3}\) :)

agentx5
 2 years ago
Best ResponseYou've already chosen the best response.1Ty for helping out on this one :D

klimenkov
 2 years ago
Best ResponseYou've already chosen the best response.0I'm really sorry to interrupt you. But it is very interesting fact, that \[\sum_{n=3}^{\infty} \frac{4}{n^21}=\lim_{n \rightarrow \infty}\left( \frac 53 \left( \frac{2}{n}+\frac{2}{n1}\right) \right) \]I enjoy this!
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