This is what my textbook says, and I see the examples that support this fact. However, I don't understand why it's true. The derivative is a linear operator. The math should work out because the Fourier Series is just a sum representation of a signal.
2026-04-06 13:11:12.1775481072
Why can't you take the derivative of a function's fourier series to get its derivative?
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A Fourier series is not merely a sum. It is an infinite sum, which it turns out is not a sum at all, but a limit of a sequence of partial sums. In particular you can't apply a linear operator to an infinite sum unless the linear operator is continuous.
You would expect the derivative to be continuous, right? Well, it's not. The easiest example I have is this:
The sequence of functions $\frac{\sin{nx}}n$ converges to $0$. The sequence of derivatives is $\cos{nx}$. For the derivative to be continuous this would have to converge to $0$, but clearly it doesn't converge to anything.