# Homework Help: Finding if a Series is Convergent

1. Apr 25, 2012

### I'm Awesome

1. The problem statement, all variables and given/known data

Does $\sum$$\frac{k}{1+k^2}$ converge?

2. Relevant equations

3. The attempt at a solution

What I'm basically doing is using the ratio test. And at the end I get that the limit goes to ∞.
Therefore it it is convergent. But I'm not sure if I'm using the right test or if an other test would be easier.

2. Apr 25, 2012

### LCKurtz

Without seeing your work, who knows where you went astray. I don't think the ratio test will help you on this problem. What other tests have you thought of trying?

3. Apr 25, 2012

### mathmajor23

If you're getting ∞ as an answer, then the series diverges, not converges.

4. Apr 26, 2012

### I'm Awesome

Okay this is what I did. I decided not to use the ratio test anymore.

http://img341.imageshack.us/img341/5567/calculus.png [Broken]

Last edited by a moderator: May 5, 2017
5. Apr 26, 2012

### LCKurtz

What you have done is pointless. Noting that the nth term of a series goes to zero tells you nothing about convergence of the series.

There are other tests for convergence/divergence than the ratio test. What are they?

Last edited by a moderator: May 5, 2017
6. Apr 26, 2012

### sid9221

I would try an Integral test. Always the best for "easy functions" cause you can't over think and make a mistake.

7. Apr 26, 2012

### I'm Awesome

Comparison test, integral test, p-series test, root and ratio tests, and the alternating series test.

I'm just not sure which on to use. I know it can't be an alternating series, or p-series.

8. Apr 26, 2012

### sharks

Use the comparison test or the limit comparison test.

Let $u_k=\frac{k}{1+k^2}$
Then, $v_k$ can be approximated to ...... as k tends towards infinity.

9. Apr 26, 2012

### Staff: Mentor

To elaborate on what LCKurtz said, the nth term test for divergence says something like this.
"If $\lim_{n \to \infty} a_n \neq 0$, the series Ʃ an diverges."

This test can tell you only whether a series diverges. It is a very common mistake that students make when they conclude from this test that a given series converges. In other words, if $\lim_{n \to \infty} a_n = 0$, you cannot conclude that the series converges.

10. Apr 26, 2012

### sharks

Example: $\lim_{n \to \infty} \frac{1}{n} = 0$ but it diverges! (it's a harmonic series).

11. Apr 26, 2012

### Staff: Mentor

Good example.

Another example is $\lim_{n \to \infty} \frac{1}{n^2} = 0$. Same result, but this time the series converges.

12. Apr 26, 2012

### sharks

Thank you for clarifying this common mistake, Mark44. I made that error not so long ago in my test. I blame it on stress.

13. Apr 26, 2012

### LCKurtz

OK. Well I have already told you the ratio test won't work and you see it isn't an alternating series. It isn't a p series. So that leaves comparison and general comparison test and integral test. So try something.

14. Apr 27, 2012

### I'm Awesome

Okay, I tried the integral test. Does that answer look better now?

Therefore, the seires is divergent by the integral test.

15. Apr 27, 2012

### Dick

Looks fine.

16. Apr 27, 2012

### Whovian

The integral test works nicely. Though what I'd do is the Ratio Test. Nice solution! Really, really nice.

17. Apr 27, 2012

### LCKurtz

Which, as I have mentioned before, would be inconclusive.

18. Apr 27, 2012

### LCKurtz

Yes. See, all you had to do was try it. Good writeup.