# Absolute convergence of the Fourier series

Given a Fourier series:

$$f(x) = \sum_{n=1}^\infty (a_n \sin(nx)+b_n \cos(nx))$$

Show it is absolutely convergent if $$\sum^\infty_{n=1}(|a_n|+|b_n|)$$ is finite.

My attempt to show it is as follows:

For $f(x)$ to be absolute convergent $$\sum_{n=1}^\infty |(a_n \sin(nx)+b_n \cos(nx))|$$ must be convergent.

Since $\sin(nx),\cos(nx)$ are bounded by $-1$ and $1$ we know $$\sum_{n=1}^\infty |a_n \sin(nx)+b_n \cos(nx)|\leq\sum_{n=1}^\infty |a_n +b_n|\leq\sum_{n=1}^\infty (|a_n| +|b_n|)$$

The last inequality is due to the triangle inequality. Now I want to conclude that if indeed $\sum^\infty_{n=1}(|a_n|+|b_n|)$ is finite then $\sum_{n=1}^\infty |a_n \sin(nx)+b_n \cos(nx)|$ converges and we have proven what we were trying to prove.

However I an not sure if my reasoning is sufficient. I would appreciate any help!

$$|a_n \sin(nx) + b_n \cos(nx)| \leq |a_n \sin(nx)| + |b_n \cos(nx)| \leq |a_n| + |b_n|.$$
In your argument, you used $|a_n \sin(nx) + b_n \cos(nx)| \leq |a_n + b_n|$ and I leave it to you to check why this inequality doesn't always hold.
• Thanks! Yeah, I realised when I was rewriting it that it would be problematic when for example $sin(nx)=-1$ but if we use the triangle inequality first it fixes the problem! – Kat.m Nov 2 '16 at 8:31