I was reading about Fourier series here when I can across the following sum: \begin{equation} \frac{a_0}{2} + \sum_{n=1}^{\infty} \frac{a_n-ib_n}{2} e^{inx} + \sum_{n=1}^{\infty} \frac{a_n + ib_n}{2} e^{-inx} = \sum_{n=-\infty}^{\infty} c_ne^{inx} \end{equation} I couldn't figure out why this was true. So I started to do the calculation myself. \begin{align*} f(x)&= \sum_{n=0}^{\infty} \left(\frac{a_n - ib_n}{2}\right) e^{inx}+ \sum_{n=0}^{\infty} \left(\frac{a_n + ib_n}{2}\right) e^{-inx}\\ &= \sum_{n=0}^{\infty} \left(\frac{ a_n - ib_n }{2}\right) e^{inx}+ \sum_{n=-\infty}^{0} \left(\frac{a_n + ib_n }{2}\right) e^{inx}\\ &= \sum_{n=-\infty}^{\infty} \left(\frac{ 2a_n + ib_n -ib_n }{2}\right) e^{inx}\\ &= \sum_{n=-\infty}^{\infty} (a_n) e^{inx}\\ \end{align*} which I could rename and thus get \begin{equation} \sum_{n=-\infty}^{\infty} c_n e^{inx} \end{equation}
My Question:
Did I use the sum sign appropriately? When I flipped it, this would make the $a_n+ib_n$ have negative index values and I wasn't sure that was allowed.
The lower indices in your summations should not start from $n = 0$; the term $a_0/2$ is separate. Really, the major mistake starts in the second line, where you reindexed the second sum to range from $-\infty$ to $0$. The coefficients $(a_n + ib_n)/2$ contained in that sum should be $$\frac{a_{\color{red}{-n}} + ib_{\color{red}{-n}}}{2}$$ So if you define coefficients $c_n$ by setting $$c_n = \begin{cases}\dfrac{a_n - ib_n}{2}, & n > 0\\\\ \dfrac{a_0}{2}, & n = 0\\\\ \dfrac{a_{-n} + i b_{-n}}{2}, & n < 0,\end{cases}$$ then $$f(x) = \sum_{n = -\infty}^\infty c_n e^{inx}.$$