Let $\sigma_N f$ be the $N$-th Cesàro mean of the Fourier series of $f$, that is, $$\sigma_N f(t) = \frac{S_0 f(t) + S_1 f(t) + \cdots + S_{N - 1} f(t)}{N}.$$ I'm showing that $$\sigma_N f(t) = \frac{a_0}{2} + \sum\limits_{k = 1}^{N - 1} \left(\left(1 - \frac{k}{N}\right)\left(a_k\cos(kt) + b_k\sin(kt)\right)\right)$$ and I wonder how to formally justify an equality:
I already got that $$\sigma_N f(t) = \frac{a_0}{2} + \frac{\sum\limits_{j = 1}^{N - 1} \sum\limits_{k = 1}^{j} \left(a_k\cos(kt) + b_k\sin(kt)\right)}{N}.$$ My problem is the following. I would like to say that $$\sum\limits_{j = 1}^{N - 1} \sum\limits_{k = 1}^{j} \left(a_k\cos(kt) + b_k\sin(kt)\right) = \sum\limits_{k = 1}^{N - 1} \left((N - k)\left(a_k\cos(kt) + b_k\sin(kt)\right)\right),$$ but how could I justify it? I know that because I thought about it by brute force… I find it hard to formalize it.