How do I show sin 1 + sin 2 + sin 3 + diverges?

[incubusfan723]

Homework Statement

I was given the series sin 1 + sin 2 + sin 3 + sin 4 + ... and I must show that it diverges. Can anyone point me in the right direction as to how to go about doing this?

Homework Equations

The Attempt at a Solution

 

[xcvxcvvc]

If the parts of a series never reach zero, wouldn't you keep adding chunks, so the sum never settles down(never converges - it diverges)?

 

[incubusfan723]

I understand it intuitively that the sum is never going to settle down due to the nature of the function, but I'm having a hard time putting that concept into a formal proof.

 

[berkeman]

If the parts of a series never reach zero, wouldn't you keep adding chunks, so the sum never settles down(never converges - it diverges)?

Note that the range of sin(x) is -1 to 1. So if you picked the arguments to the sin(x) function correctly, you could get the sum to not diverge.

The hard part of this qestion is that the x values are not an even fraction of PI...

 

[chroot]

http://en.wikipedia.org/wiki/Convergence_tests

The first rule should suffice.

- Warren

 

[berkeman]

http://en.wikipedia.org/wiki/Convergence_tests

The first rule should suffice.

- Warren

Looks like the first rule only applies for summands > 0. Good link, though. Time for some reading...

 

[berkeman]

Looks like the first rule only applies for summands > 0. Good link, though. Time for some reading...

Or more accurately, the first rule is talking about converging, as opposed to not diverging, which the OP is asking about.

 

[chroot]

No, berkeman. The summand in this case is sin(n), and the limit of sin(n) as n goes to infinity is undefined. Therefore, the series does not converge.

- Warren

 

[berkeman]

No, berkeman. The summand in this case is sin(n), and the limit of sin(n) as n goes to infinity is undefined. Therefore, the series does not converge.

- Warren

I totally agree it does not converge. But the OP said the problem was to show that it diverges. The sum can oscillate without diverging.

I just did a quick Excel check to get a feel for what the sum does. After 100 terms, it's not obvious that it will diverge...

 

[xcvxcvvc]

http://en.wikipedia.org/wiki/Convergence_tests

The first rule should suffice.

- Warren

this is the right answer. to show it, take the limit of sin(x) as x approaches infinity (which is undefined)

Note that the range of sin(x) is -1 to 1. So if you picked the arguments to the sin(x) function correctly, you could get the sum to not diverge.

The hard part of this qestion is that the x values are not an even fraction of PI...

the only way an infinite series of sine could converge is if each piece equaled zero. for example, the series of sin(k) when k = pi and k increased by 2pi each time. (0 + 0 + 0... adds up to 0)

 

[Mark44]

What they refer to in the wiki article as "Limit of the summand" is similar to one that appears in some calculus texts, the Nth Term Test for Divergence. I.e., in a series $\sum a_n$, if lim a_n is not zero, the series diverges.

A common mistake that students make happens when they find that lim a_n = 0, and conclude that $\sum a_n$ converges.

 

[berkeman]

After 500 terms, the running series sum is spending more time postive than negative, but the highest it's getting is just under 2.

 

[xcvxcvvc]

What they refer to in the wiki article as "Limit of the summand" is similar to one that appears in some calculus texts, the Nth Term Test for Divergence. I.e., in a series $\sum a_n$, if lim a_n is not zero, the series diverges.

A common mistake that students make happens when they find that lim a_n = 0, and conclude that $\sum a_n$ converges.

yeah, lol. i remember people confused about the topic. i just giggled inside.


if i am mad, my face is red.

i am not mad, so my face must not be red :D

wait, what about when a cute girl walks my way

 

[robphy]

Can an Euler-type sum help?

S_n=sin 1 + sin 2 + sin 3 + sin 4 + ... +sin n
S_n=sin n + sin (n-1) + sin (n-2) + sin (n-3) + ... +sin 1

2S_n= (sin 1 + sin n) + (sin 2 + sin (n-1) ) +... + (sin n + sin 1)

Then, use sin a+sin b identities to get a closed form expression.
Study n -> infty.

 

[rwisz]

I would just do what's already been mentioned and write the series as

$$\sum_{n=1}^{\infty}sin(n)$$

then just perform the n^th test for divergence to show that the limit as n approaches infinity of sin(n) is infinity, and therefore NOT zero, and therefore it DIVERGES. And as otherwise mentioned, intuitively you know that sine from 1 to infinity will always oscillate from -1 to 1, and thus never converge to a single value.

Unless you have to write the proofs for the nth test for divergence... I'm not really sure what else there is to be done here...

 

[chroot]

The sum can oscillate without diverging.

At least in the mathematical sense, oscillation is a form of divergence, because the series sum never settles down to any specific value. The nth term test is all that's needed to show that this series diverges. I've even heard this called the "hurdle test" by math teachers, because if it doesn't pass this hurdle, it cannot converge, and there's no need to check anything else.

- Warren

 

[Mark44]

At least in the mathematical sense, oscillation is a form of divergence, because the series sum never settles down to any specific value. The nth term test is all that's needed to show that this series diverges. I've even heard this called the "hurdle test" by math teachers, because if it doesn't pass this hurdle, it cannot converge, and there's no need to check anything else.

- Warren

I agree completely. A series doesn't have to have sums that become ever larger or ever more negative to diverge.

 

[MeTh0Dz]

Perhaps you could prove divergence by the "Grouping Theorem" as done with the Harmonic Series here -> http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Alternate_proof_of_divergence.

 

[Count Iblis]

You can also evaluate:

$$S(N)=\sum_{n=0}^{N}\sin(n) = \text{Im}\sum_{n=0}^{N}\exp(i n)=\text{Im}\frac{1-\exp[i(N+1)]}{1-\exp(i)}$$

So,

S(N) = [sin(1) + sin(N) - sin(N+1)]/[2 - 2 cos(1)]

and we see that lim N to infinity of S(N) does not exist.