Posted by **Alanna** on Thursday, May 31, 2012 at 8:49pm.

Show that the series cos(n) from n=1 to infinity is divergent.

- Math -
**MathMate**, Friday, June 1, 2012 at 12:15pm
The sum of the series ∑cos(nx) is, according to Mathworld,

N

∑cos(nx) = cos(Nx/2)sin((N+1)x/2) / sin(x/2)

n=0

(Note that the summation starts from 0, make adjustments accordingly).

The given series has x=1, or

S(1)=∑cos(n)

We see that the sum to N oscillates as N increases. Since we cannot find a value of N for whcih |T(n+1)|/|T(n)|<1 ∀n>N , we conclude that the series does not converge.

I make a difference between divergence where the sum approaches ±∞ and where the sum oscillates. I call the latter non-convergent.

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