The partial sums S_N(f; x) are defined as \[ S_n (f ; x) = \frac{a_0}{2} + \sum_{k= 1}^n \left( a_k \,\cos \left( k\,\frac{\pi}{\ell}\,x \right) + b_k \,\sin \left( k\,\frac{\pi}{\ell}\,x \right) \right) . \] Since the sequence Sₙ(f; x) converges uniformly to f(x), it follows that \( \displaystyle \quad S_k (x)\,\cos \left( n\,\frac{\pi}{\ell}\,x \right) \quad \) converges uniformly to f(x) cos(nπx/ℓ) as k → ∞ for each fixed n. Observe that |Sₖ(x) cos(nπx/ℓ) − f(x) cos(nπx/ℓ)| ≤ |Sₖ(x) − f(x)|. Therefore, for each fixed n, \[ f(x)\,\cos \left( n\,\frac{\pi}{\ell}\,x \right) = \frac{a_0}{2}\,\cos \left( n\,\frac{\pi}{\ell}\,x \right) +\sum_{k\ge 1} \left( a_k \,\cos \left( k\,\frac{\pi}{\ell}\,x \right) + b_k \,\sin \left( k\,\frac{\pi}{\ell}\,x \right) \right) \cos \left( n\,\frac{\pi}{\ell}\,x \right) \] The uniformly convergent series may be integrated term by term \[ \int_{-\ell}^{\ell} f(x)\, \cos \left( n\,\frac{\pi}{\ell}\,x \right) {\text d}x = \ell\,a_n , \qquad n=0,1,2,\ldots . \] The argument goes analog for product of f(x) and sine function.