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!


Your reasoning is sufficient as the convergence of the left hand side follows from the convergence of the right hand side by the comparison test. The only problem in your argument is that you should use the triangle inequality before as in

$$ |a_n \sin(nx) + b_n \cos(nx)| \leq |a_n \sin(nx)| + |b_n \cos(nx)| \leq |a_n| + |b_n|. $$

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.

  • $\begingroup$ 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! $\endgroup$ – Kat.m Nov 2 '16 at 8:31

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