0

I think the series $\sum_{k=0}^{\infty}\vert\cos(ak)\vert$, where $a$ is a non-zero constant, is absolutely convergent. But I had a trouble proving it. Can anyone give me a hint on how to prove it? Thanks!

Gregory Grant
  • 14,874
SemiMetrics
  • 61
  • 1
    It definitely isn't. If you take $a$ such that there is one $k$ so $ka=\pi$, then you get infinitely many $k$ with $\cos{ak}=1$, so the series has infinitely many $1$s in it. – Chappers May 04 '15 at 16:04

1 Answers1

1

This is not true. For instance, look at what happens when $a=\pi/2$. We have $$\limsup_n \vert \cos\left(n\pi/2 \right)\vert = 1$$

In fact this is not true for any $a$, since we can prove that for any $a$, we have $$\limsup_n \vert \cos\left(na \right)\vert = 1$$

Adhvaitha
  • 20,259