Is the Series ∑ (sin(1/n)/√n) Convergent?

[hpayandah]

Homework Statement

Ʃ^{∞}_{n=1} \frac{sin(1/n}{\sqrt{n}}

Homework Equations

The Attempt at a Solution

lim_{n→∞} \frac{sin(1/n}{\sqrt{n}}= lim_{n→∞} \frac{-2cos(1/n)}{n^{3/2}} with l'hospital rule = 0

since lim_{n→∞}=0 therefore the series is convergent

Do you think I did this right?

 

[HallsofIvy]

I can't imagine why you would use L'Hopital for that: sin(2/n) lies between -1 and 1 for all n while the numerator gets larger and larger. Of course, the limit is 0.


But that doesn't help you. a_n going to 0 is a necessary condition for convergence of the series \sum a_n, it is not a sufficient condition.

If a_n does NOT go to 0, then \sum a_n does not converge but there exist series in which a_n converges to 0 but the series \sum a_n does not converge.

For example, \sum 1/n clearly has 1/n going to 0 but the series does not converge.

 

[Yuqing]

I can't imagine why you would use L'Hopital for that: sin(2/n) lies between -1 and 1 for all n while the numerator gets larger and larger. Of course, the limit is 0.


But that doesn't help you. a_n going to 0 is a necessary condition for convergence of the series \sum a_n, it is not a sufficient condition.

If a_n does NOT go to 0, then \sum a_n does not converge but there exist series in which a_n converges to 0 but the series \sum a_n does not converge.

For example, \sum 1/n clearly has 1/n going to 0 but the series does not converge.

Consider doing a limit comparison test. What well known limit involving sine might help here?

 

[hpayandah]

Thank you for quick reply. Okay now I understand that the proof is not sufficient. Can I instead use the fact that you said -1 < sin(1/n) < 1 and the graph shows this:

Graph attached.

can I say since \frac{sin(1/n}{\sqrt{n}} < \frac{1}{\sqrt{n}} then use the p-series to show convergence or divergence.
or I can think of continuing what I was doing and prove that n^(3/2) converges then by definition -2con(1/n) should also converge.

 

[Ray Vickson]

Thank you for quick reply. Okay now I understand that the proof is not sufficient. Can I instead use the fact that you said -1 < sin(1/n) < 1 and the graph shows this:

Graph attached.

can I say since \frac{sin(1/n}{\sqrt{n}} < \frac{1}{\sqrt{n}} then use the p-series to show convergence or divergence.
or I can think of continuing what I was doing and prove that n^(3/2) converges then by definition -2con(1/n) should also converge.

You are doing way too much work. When n is large, 1/n is small and sin(1/n) is 1/n + O(1/n^2), so the nth term is 1/n^(3/2) + O(1/n^(5/2)). Convergence is then obvious.

RGV