Ace school

with brainly

  • Get help from millions of students
  • Learn from experts with step-by-step explanations
  • Level-up by helping others

A community for students.

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...

Mathematics
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Join Brainly to access

this expert answer

SIGN UP FOR FREE
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.
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?
\[\sum_{n=3}^{\infty} \frac{-2}{n+1} + \sum_{n=3}^{\infty} \frac{2}{n-1}\]

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

\[\sum_{n=3}^\infty \left(\frac{2}{n-1}-\frac{2}{n+1}\right)\]seems like a telescoping series to me.
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
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.
Ah ok, that makes more sense because http://www.wolframalpha.com/input/?i=telescoping+series&lk=4 wasn't helping :P
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.
+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?
Oh wait I think I may have misunderstood, the first few that don't cancel out are what will be left
Yes?
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.
Wonderful information, I'm evaluating it by hand at the moment. I'll post what I've found when I'm done. :-)
I got \(\large\frac{5}{3}\) :-)
That looks right.
1 + 2/3 = 5/3
Ty for helping out on this one :-D
You're welcome.
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!

Not the answer you are looking for?

Search for more explanations.

Ask your own question