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Dirichlet Test

by hth
Tags: dirichlet, test
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hth
#1
Dec15-09, 06:15 PM
P: 26
1. The problem statement, all variables and given/known data

Given the infinite series, [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.

2. Relevant equations

3. The attempt at a solution

Let f_n(x) = sin(nx) and g_n(x) = 1/n^(s)

i.) lim n-> [tex]\infty[/tex] f_n = 0

I'm not sure how to show this formally. Specifically, for a sequence instead of a function.

ii.) [tex]\sum[/tex] |g_(n+1) - g_n| converges.

Or this either.

iii.) [tex]\sum[/tex] g_n, its partial sums are uniformly bounded.

[tex]\sum[/tex] |g_(n+1) - g_n| = (g_1 - g_2) + (g_2 - g_3) + .... + (g_n - g_(n+1) = g_1 - g_(n+1). Lim n->infinity [tex]\sum[/tex] |g_(n+1) - g_n|= lim n->infinity (g_1 - g_(n+1)) = g_1. Therefore the partial sums are bounded, so [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.
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Mark44
#2
Dec16-09, 12:53 AM
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P: 21,311
Quote Quote by hth View Post
1. The problem statement, all variables and given/known data

Given the infinite series, [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.
Can you give us the problem verbatim? What you have written is not as clear as it should be. Are you supposed to prove that your series converges if 0 < s <= 1?

Unless I am mistaken, this will be very difficult to prove, because it isn't true.
Quote Quote by hth View Post

2. Relevant equations

3. The attempt at a solution

Let f_n(x) = sin(nx) and g_n(x) = 1/n^(s)

i.) lim n-> [tex]\infty[/tex] f_n = 0
This is not true. For an arbitrary value of x, the sequence f_n(x) does not converge. The values lie in a band between -1 and 1.
Quote Quote by hth View Post

I'm not sure how to show this formally. Specifically, for a sequence instead of a function.

ii.) [tex]\sum[/tex] |g_(n+1) - g_n| converges.
This might be true, but it doesn't have anything to do with this problem that I can see.
Quote Quote by hth View Post

Or this either.

iii.) [tex]\sum[/tex] g_n, its partial sums are uniformly bounded.
The partial sums would be the sequence
[tex]S_k~=~\sum_{n = 1}^k \frac{1}{n^s} [/tex]
Quote Quote by hth View Post
[tex]\sum[/tex] |g_(n+1) - g_n| = (g_1 - g_2) + (g_2 - g_3) + .... + (g_n - g_(n+1) = g_1 - g_(n+1). Lim n->infinity [tex]\sum[/tex] |g_(n+1) - g_n|= lim n->infinity (g_1 - g_(n+1)) = g_1. Therefore the partial sums are bounded, so [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.
hth
#3
Dec16-09, 07:13 AM
P: 26
Hi, thank you for replying. Here's a re-wording of the problem.

Use the Dirichlet test to show that the infinite series from n=1 to infinity of sin(nx)/n^(s) is convergent for 0 less than s less than or equal to 1. Note: x is any real number.

Mark44
#4
Dec16-09, 08:56 AM
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P: 21,311
Dirichlet Test

The Dirichlet test applies to series of the form
[tex]~\sum_{n = 1}^{\infty} a_nb_n [/tex]
To use this test you need to show that the following are true:
  1. an >= an+1 > 0, with an = 1/ns (The other sequence, sin(nx), is not a decreasing, bounded sequence.)
  2. lim an = 0, as n --> [itex]\infty[/itex]
  3. [tex]\left|\sum_{n = 1}^N b_n \right| \leq M~for~all~N[/tex], and where M is a positive constant

The first two items are pretty easy to show, but the third requires some work.
hth
#5
Dec16-09, 10:59 AM
P: 26
Alright, so,

So,

i) Let (n+1)^(s) = n^s + sC1 x^(s-1)(1) + sC2 x^(s-2)(1)^2 +

.....

Now, assuming n ≥ 0 we get that n^s is a small part or 1st term of

the right hand side of above expression is ≥ the left hand side.

Note: The other part sC1x^(s-1)(1) + sC2 x^(s-2)(1)^2 + ... is

positive and subtracted to get n^s or n^s ≥ (n+1)^(s)

Then, taking the reciprocal we change signs and rearranging we get

1/n^(s) ≥ 1/(n+1)^(s).

ii) Let be |n| = 1+δ, δ>0.

Let be S so that δ>1/S

Therefore

|n^s| > (1+1/S)^s = (S+1)^s/S^s =
= (S^s + s S^(s-1) + ...)/S^s >
> s/S

If s>SM, then

|n^s| > M

and then

for each ε>0 there is M > 1/ε and for every s > S_o = SM,

|1/n^s| < 1/M < ε

Then, lim 1/n^s = 0.

iii) I don't really know where to start here, honestly.
Mark44
#6
Dec16-09, 01:05 PM
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P: 21,311
No. Look at what I wrote in post #4. Steps 1 and 2 have to do with an, which I have identified as 1/ns. Those steps are pretty easy to show.

Step 3 deals with bn, which I am identifying as sin(nx). You have to show that there is a positive constant M for which the following inequality is true for all N and any x.
[tex]\left|\sum_{n = 1}^N sin(nx) \right| \leq M[/tex]
hth
#7
Dec16-09, 02:08 PM
P: 26
What did I do wrong in steps 1 & 2?
Mark44
#8
Dec16-09, 03:05 PM
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P: 21,311
I glossed over what you were doing and misunderstood what you were saying. In any case, to show steps 1 and 2 for a_n = 1/n^s, you can probably say what needs to be said in less than a quarter of what you said. You really don't need to expand (n + 1)^2, and you don't need to proved via epsilons and deltas that 1/n^2 --> 0.
hth
#9
Dec16-09, 03:06 PM
P: 26
Alright, here's my attempt at part 3.

[tex]\sum[/tex] sin(nx) = [(cos (x/2) - cos(n + 1/2)x) / (2sin(x/2))] for all x with sin(/2) =/= 0.

We have,

sin(x/2) [tex]\sum[/tex] sin(nx) = sin(x/2) sinx + sin(x/2) sin 2x + ... + sin (x/2) sin(n)

By the identity, 2 sin A sin B = cos (B-A) - cos (B + A), this becomes,

2sin(x/2) [tex]\sum[/tex] sin(nx) = cos (x/2) - cos (n + 1/2)x.

So, the partial sums are bounded by 1/|sin(x/2)|. Therefore, [tex]\sum[/tex] sin(nx)/n^(s) converges.
Mark44
#10
Dec16-09, 04:33 PM
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P: 21,311
This looks like a good start at it. Some comments along the way.
Quote Quote by hth View Post
Alright, here's my attempt at part 3.
What you have below is hard to follow, since it appears to be the conclusion from the work below it. You should give the reader a clue as to where it comes from.

I think this is what you mean to say.
[tex]\sum_{n = 1}^N sin(nx)~=~\frac{cos (x/2) - cos((N + 1/2)x)}{2sin(x/2)} [/tex]
for all x such that sin(x/2) =!= 0.
Quote Quote by hth View Post
[tex]\sum[/tex] sin(nx) = [(cos (x/2) - cos(n + 1/2)x) / (2sin(x/2))] for all x with sin(/2) =/= 0.

We have,
Instead of saying "We have," you really mean "because."
Quote Quote by hth View Post
sin(x/2) [tex]\sum[/tex] sin(nx) = sin(x/2) sinx + sin(x/2) sin 2x + ... + sin (x/2) sin(n)

By the identity, 2 sin A sin B = cos (B-A) - cos (B + A), this becomes,

2sin(x/2) [tex]\sum[/tex] sin(nx) = cos (x/2) - cos (n + 1/2)x.

So, the partial sums are bounded by 1/|sin(x/2)|. Therefore, [tex]\sum[/tex] sin(nx)/n^(s) converges.
Now you've lost me. You have
[tex]\sum_{n = 1}^N sin(nx)~=~\frac{cos (x/2) - cos((N + 1/2)x)}{2sin(x/2)} [/tex]

so how do you conclude that this partial sum is bounded by 1/|sin(x/2)|?
hth
#11
Dec16-09, 04:51 PM
P: 26
I left out some work, let me pick back up here:

By the identity, 2 sin A sin B = cos (B-A) - cos (B + A), this becomes,

2sin(x/2)[tex]\sum[/tex] from 1 to n of sin kx = (cos x/2 - cos 3x/2) + (cos 3x/2 + cos 5x/2) + ... + (cos (n-1/2)x - cos (n+1/2)x)

= cos (x/2) - cos (n + 1/2)x.
Mark44
#12
Dec16-09, 06:16 PM
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P: 21,311
I understand that. My question is how do you conclude that this partial sum is bounded by 1/|sin(x/2)|?
hth
#13
Dec16-09, 07:34 PM
P: 26
Wait, lets go back to the second part here.

a_n >= a_n+1 > 0, with an = 1/ns

Shouldn't that be ∑|a_(n+1) - a_(n)| converges instead??
Mark44
#14
Dec16-09, 08:01 PM
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P: 21,311
No, not at all. To use the Dirichlet test (which is used on a summation of the product of elements of two sequences), one of the sequences has to be decreasing (a_n >= a_n+1) and bounded below by zero. That's the same as saying this sequence converges to zero. Where are you getting this: ∑|a_(n+1) - a_(n)| ?
tracedinair
#15
Dec16-09, 08:49 PM
P: 50
I got it from my textbook, actually, haha.
Dick
#16
Dec16-09, 11:52 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
Quote Quote by Mark44 View Post
Instead of saying "We have," you really mean "because."
Now you've lost me. You have
[tex]\sum_{n = 1}^N sin(nx)~=~\frac{cos (x/2) - cos((N + 1/2)x)}{2sin(x/2)} [/tex]

so how do you conclude that this partial sum is bounded by 1/|sin(x/2)|?
Not to intrude, but the numerator is bounded by 2, right? I think it is bounded by 1/|sin(x/2)|.
Mark44
#17
Dec17-09, 12:20 AM
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P: 21,311
That works for me. Thanks for jumping in, Dick. I don't feel intruded on at all.


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