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Calculus/Infinite Series

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Expert: - 2/23/2010


Question
How would I find whether the summation of arctan(k)/(1+k^2) from 1 to infinity diverges or converges?
I think I should use the Limit Convergence test or the Comparison test, but I'm not sure how to go about it.
Thanks!

Answer
It is known that the summation of 1/kČ converges.

Each term in the summation of 1/(1+kČ) is less than this,
so this series converges as well.

When multiplied by arctan(k), the results of arctan(x) are inclusively between 0 and 2π.
Since it is known that 1/kČ converges, it is also known that 2π/kČ converges.

So each term of the series 2π/kČ can be shown to be greater than each term of arctan(k)/(1+kČ), and the the 1st series is known to converge, so the 2nd series must also converge.

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