Show series [sin(n)]/n converges?

[DavidWhitbeck]

That's a very interesting question. I agree Dick's method is very nice and seems to work. However I also agree that the techniqes used might be too advanced.
A proof using only elementary comparison tests and the like would certainly be much more technical and I see no reason why it should be only a few lines long. I did something similar some time ago maybe I can find my notes.

I said that it should be only a few lines long because otherwise it's not an acceptable problem for a high school calculus class.

And the reason it should involve comparison tests is because that's what they're taught in the class (standard topic for Calc BC which the OP said was taking). The problem must certainly be an exercise in the tools that they were taught. In a class like that, they're not going to deal with fun problems that require tricks, which is a pity.

Also comparison tests make up the last topic in a BC course, and it's April, if a student is posting a question now it's either from reviewing AP exams or because they are still studying comparison tests!

 

[Dick]

That's a very interesting question. I agree Dick's method is very nice and seems to work. However I also agree that the techniqes used might be too advanced.
A proof using only elementary comparison tests and the like would certainly be much more technical and I see no reason why it should be only a few lines long. I did something similar some time ago maybe I can find my notes.

I agree with you guys. My 'method' isn't even a proof. It involves manipulating divergent series. I think a genuine proof would involve real analysis techniques, e.g. you are definitely going to need that pi is irrational, and that's not an elementary proof. When the instructor said the problem was difficult but they could do it the only thing simple enough I could think of was deMoivre. I'd LOVE to see this elementary proof. Find those notes Pere Callahan!

 

[DavidWhitbeck]

Yeah find those notes! I'm going to ask a couple of people that would know better than me-- my students. They are taking Calc BC now, and they get 5's on every test. Since it's fresh on their minds I bet they could figure it out real fast, while I'm still stuck scratching my head trying to remember what it was all about.

 

[lurflurf]

try sumation by parts
ΣuΔv=uv-ΣΔuEv
btw the sum is (π-1)/2

 

[drpizza]

How about writing the numerator as a MacLaurin Series, then dividing out the n in the denominator? Then, use a comparison test and compare it to the MacLaurin Series for Cos(n)?

\sin{n} = n - \frac{n^3}{3!} +\frac{n^5}{5!} - \frac{n^7}{7!} +\frac{n^9}{9!} . . .

Therefore,

\Sigma\frac{\sin{n}}{n} = 1 -\frac{n^2}{3!} +\frac{n^4}{5!} - \frac{n^6}{7!} +\frac{n^8}{9!} . . .

Compare that to

\cos{n} = 1 - \frac{n^2}{2!} +\frac{n^4}{4!} - \frac{n^6}{6!} +\frac{n^8}{8!} . . .

 

[lurflurf]

How about writing the numerator as a MacLaurin Series, then dividing out the n in the denominator? Then, use a comparison test and compare it to the MacLaurin Series for Cos(n)?

Oh my you have lost the sum. Thus you have introduced a double sum. Analysis of such a some is likely harder than that of the original sum.

 

[lurflurf]

Do you know how to find Fourier expansions?
consider
(pi-x)/2 0<x<2pi
then consider x=1

 

[Dick]

try sumation by parts
ΣuΔv=uv-ΣΔuEv
btw the sum is (π-1)/2

Hi lurflurf,

You obviously know what you are doing. I was writing the answer as arctan(sin(1)/(1-cos(1))), but yeah, that is more cleanly expressed as (pi-1)/2. But how do you set this up using this 'summation by parts'? And I haven't figured out the how the Fourier expansion works yet, but I'll look at it tomorrow. Really nice work. Most other contributions to this thread have been completely clueless.

 

[Gib Z]

I haven't ever seen this summation by parts formula before, though I should have expected it exists, there must have been some discrete form for the continuous counterpart. As for the Fourier expansion, that seems even more out of such a course, though It is also a nice solution =]

Dick; as for the Fourier expansion, since f(x) = x is an odd function, the a coefficient is automatically zero whilst b_v = \frac{2}{\pi} \int^{\pi}_0 x \sin vx dx which some quick integration by parts yields b_v = (-1)^{v+1} \frac{2}{v}, and so we get x = 2 \left( \frac{\sin x}{1} - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} + ... \right).

Rearranging and letting x=1 gives us the desired result, and putting in x= pi/2 gives us a very famous result =]

EDIT: O god i just realized I found the wrong Fourier series. argh

 

[Dick]

Hmm. If you did everything right then x=1 does not give the desired solution. That's an alternating sign series (-1)^n*sin(n)/n. The original question doesn't have the alternating sign. Is there an extra sign that makes it work? But like I said, I didn't really look at that yet. I was trying to figure out the summation by parts thing. It makes perfect sense it such a thing would exist though. Thanks, Gib.

 

[Dick]

I haven't ever seen this summation by parts formula before, though I should have expected it exists, there must have been some discrete form for the continuous counterpart. As for the Fourier expansion, that seems even more out of such a course, though It is also a nice solution =]

Dick; as for the Fourier expansion, since f(x) = x is an odd function, the a coefficient is automatically zero whilst b_v = \frac{2}{\pi} \int^{\pi}_0 x \sin vx dx which some quick integration by parts yields b_v = (-1)^{v+1} \frac{2}{v}, and so we get x = 2 \left( \frac{\sin x}{1} - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} + ... \right).

Rearranging and letting x=1 gives us the desired result, and putting in x= pi/2 gives us a very famous result =]

EDIT: O god i just realized I found the wrong Fourier series. argh

Ok, if you do the right Fourier series, it does work. And yep, putting in x=pi/2 gives you the Gregory-Leibniz formula for pi. Interesting.

 

[drpizza]

Oh my you have lost the sum. Thus you have introduced a double sum. Analysis of such a some is likely harder than that of the original sum.

Ooops! That was as silly mistake. Should have worked it on paper first - I lose things when I'm typing them sometimes.

 

[Dick]

For future reference, a systematic approach to this uses the Abel-Dedekind-Dirichlet theorem. The product of a sequence with bounded partial sums (sin(n), use a trig identity) and a function of bounded variation (1/n), converges as a series. This is proved using Abel's 'summation by parts', as lurflurf intimated.

 

[DavidWhitbeck]

For future reference, a systematic approach to this uses the Abel-Dedekind-Dirichlet theorem. The product of a sequence with bounded partial sums (sin(n), use a trig identity) and a function of bounded variation (1/n), converges as a series. This is proved using Abel's 'summation by parts', as lurflurf intimated.

I came across that theorem yesterday and I was seeing if it fit with that problem, but I don't think it does because the partial sums of sin(n) must be bounded. How does one show that those partial sums are bounded?

 

[Dick]

Use sin(nx)=[cos((n-1/2)x)-cos(n+1/2)x)]/(2*sin(x/2)). It's a telescoping series. So if A_n is the partial sum, |A_n|<=1/|sin(x/2)|. Clever, huh? Wish I'd thought of it. It shows things like sin(2n)/n converge also.

 

[DavidWhitbeck]

Use sin(nx)=[cos((n-1/2)x)-cos(n+1/2)x)]/(2*sin(x/2)). It's a telescoping series. So if A_n is the partial sum, |A_n|<=1/|sin(x/2)|. Clever, huh? Wish I'd thought of it. It shows things like sin(2n)/n converge also.

That is very clever!

Well that problem would be appropriate for upper level math (analysis for use of the Dirichlet test, or applied math for engineers/physicists for recognizing that it's a Fourier series). But I think it's now safe to say that was probably an inappropriate problem for freshman calculus.

 

[Dick]

That is very clever!

Well that problem would be appropriate for upper level math (analysis for use of the Dirichlet test, or applied math for engineers/physicists for recognizing that it's a Fourier series). But I think it's now safe to say that was probably an inappropriate problem for freshman calculus.

Probably so, I think the student who asked this ran away long ago.

 

[NateTG]

When I look at something like that, my temptation is to try to find an upper bound for:
f(n)=\sum_{j=1}^{k} \sin(jx)
using geometry, and then to rearrange things into a sum that should look like:
\sum \frac{1}{n^2+n} f(n)

 

[Ratio Test =)]

\sum_{n=1}^{\infty} a_n \, sin(n)
converges whenever {an} is a decreasing sequence that tends to zero.

By : "[URL Black"]Dirichlet's test[/B][/U][/URL]

 

[myriadbeings]

Dirichlet's test will work for this series.
we can think of sin(n)/n as the product sin(n)*1/n.
dirichlet's test says that if one of the sequences in the product is bounded, and the other is monotone and converges to 0 then the series of the product of sequences must converge.
the sequence of partial sums of sin(n) is bounded and certainly 1/n is monotone and converges to 0, so the series must converge.

 

[wisvuze]

nice discussion here; I know the thread is a bit old but since so much trouble was stirred over this, I would like to also point out that dirichlet's test would have worked with this ( and would have proven this fairly easily ). It uses a lemma by Abel which says that if {bn} is non-decreasing and non-negative for each n, and terms a1,..,an are bounded: m <= a1 +... + an <= M for any n , then bk*m <= a1b1 +... + anbn <= bk*M.

Development of these theorems can be found in Spivak's Calculus book on chapters 22 (chapter on infinite series ) and 19 (chapter on integration in elementary terms ). In chapter 22, the Dirichlet test is developed in exercise 13 and in chapter 19, abel's theorem is developed in problem 35.
Not saying that it is very difficult at all, but for anyone who may be curious -- those are excellent sources

 

[myriadbeings]

one other technique that would work is an interesting criterion involving the partial sums. let sn be the nth partial sum of the series. If for every epsilon greater than zero there exists an N such that for all n>N we have
|sn+k - sn|<epsilon for all k >= 1, then the series must converge. (Knopp "Theory and Application of Infinite Series") A quick induction on k would make quick work of this series' convergence.