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

  • Math -

    The sum of the series ∑cos(nx) is, according to Mathworld,
    ∑cos(nx) = cos(Nx/2)sin((N+1)x/2) / sin(x/2)
    (Note that the summation starts from 0, make adjustments accordingly).
    The given series has x=1, or

    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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