Finding if a Series is Convergent

[I'm Awesome]

Homework Statement

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

Homework Equations

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.

 

[LCKurtz]

Homework Statement

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

Homework Equations

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.

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?

 

[mathmajor23]

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

 

[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

 

[LCKurtz]

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

http://img341.imageshack.us/img341/5567/calculus.png

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?

 

[sid9221]

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

 

[I'm Awesome]

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?

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.

 

[DryRun]

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.

 

[Mark44]

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.

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 a_n 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.

 

[DryRun]

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 a_n 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.

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

 

[Mark44]

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

Good example.

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

 

[DryRun]

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

 

[LCKurtz]

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.

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.

 

[I'm Awesome]

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

Therefore, the seires is divergent by the integral test.

 

[Dick]

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

Therefore, the seires is divergent by the integral test.

Looks fine.

 

[Whovian]

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

 

[LCKurtz]

The integral test works nicely. Though what I'd do is the Ratio Test.

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

 

[LCKurtz]

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

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