To what value do these series converge?

[defunc]

Consider these series:
sin(n)/n; and
cos(n)/n.
To what value do they converge? You can use Diriclet's test to show that they do converge, but to what? I think I should use a Fourier trigonometric series, but not certain how.

 

[arithmetix]

Without ado, the answer must be "zero", because sin(n) and cos(n) can only have values between -1 to +1, while n tends to infinity.

 

[defunc]

I meant series, not sequence. With summation n = 1 to infinity. Sorry for any confusion.

 

[l'Hôpital]

Note: I'm not sure if this actually works, but it's an idea.

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$$\sum$$ cos kx = -.5 + (.5sin[(2n +1)x/2)] )/sin x/2

So, if you integrate the sum, from 0 to 1, you should get

$$\sum$$ (sin k)/k , which is the sum you want, so integrate the other side from 0 to 1.

 

[defunc]

Thanks for the idea. I am having a hard time to find a way to integrate that expression. Meantime I was able to calculate the series sin(n)/n by differentiating the fouries cosine series cos(nx)/n^2 and substituting 1. It equates to pi/2-1
/2. I am still stuck trying to evaluate cos(n)/n however.

 

[g_edgar]

First evaluate $\sum z^n/n$ then substitute in that to evaluate $\sum e^{int}/n$, then take real and imaginary parts to get $\sum \cos(nt)/n$ and $\sum \sin(nt)/n$