Question

How can I use a p-series comparison to show whether the sum of arctan(2n)/ n^2 from n=1 to n= infinite is convergent or divergent?

Answer 1

The numerator (arctan(2n)) is bounded by pi/2, so we can say\[\sum_{k=1}^{\infty} \frac{\arctan(2k)}{k^2}\le \sum_{k=1}^{\infty} \frac{\pi/2}{k^2}<\infty\]

Answer 2

I suppose you could put another step in if you want, pulling the constant (pi/2) out in front of the sum for illustrative purposes, but there isn't any mathematical requirement to do so.

Answer 3

Ok, thank you very much!

Answer 4

No sweat. Happy to help.