# Value that this series converges to

1. May 12, 2012

Ʃ4/(n^2-1) n=2

Need help trying to solve it. Ive used other tests like the ratio test and they have all resulted inconclusive.

It does converge because the Lim as n→∞ equals 0. Do not know where to go from here. Some help would be nice! Thank you!

2. May 12, 2012

### gopher_p

Could you tell me the name of the test that gives this conclusion?

3. May 12, 2012

it was a test for divergence, i was told if the lim n→∞ of an ≠ 0 then it diverges

4. May 12, 2012

### gopher_p

Does this test ever tell you that a series converges?

5. May 12, 2012

not explicitly, but if it does not diverge then i am assuming it must converge

6. May 12, 2012

### gopher_p

Well it's a common misconception, so don't feel bad. But the test for divergence is not always conclusive (in fact it often isn't). It only singles out series that don't meet the minimum requirement for convergence. It's kind of like an entrance exam; you gotta pass it to get in, but it's no guarantee that you're gonna be successful.

So bottom line, the test for divergence only tells you that a series diverges. It never tells you when a series doesn't diverge (i.e. it never tells you that a series is convergent).

Look at the harmonic series for a stock counterexample to the kind of reasoning you tried to use.

7. May 12, 2012

### gopher_p

As far as your problem goes, if I tell you that you've got yourself a telescoping series, is that enough to get you going?

8. May 12, 2012

yes actually, thank you! i just needed a direction :)

9. May 12, 2012

### gopher_p

Good.

Now as far as having a good strategy heading into these kinds of problems (so this isn't about math, really; it's about problems you run across in math classes):

There are basically only two elementary ways that we can find the values of convergent series (remember the tests only tell you, when they're conclusive, whether or not a series converges/diverges. they don't tell you what it converges to). (1) Compute the limit of the partial sums and (2) get clever with some tricks with Power Series (don't worry if you haven't seen them). The kinds of series whose partial sums we can compute so that we can take a limit are essentially the geometric series (for which we have a formula) and telescoping series.

So if you have a problem that asks you to compute the value of a convergent series, there's a real good chance that it's either a geometric series or a telescoping series. Geometric series are relatively easy to identify, so ...

10. May 12, 2012

### Ray Vickson

$$\sum \frac{4}{n^2-1} n = \sum \frac{4n}{n^2-1}, \text{ or is it } \sum \frac{4}{(n^2-1)n}?$$