When taking the derivative of an infinite sum, do you simply differentiate the inner function and take the limit as the iterative variable asymptotically approaches infinity, or is it more complex? In other words, is this a true statement?
$$\frac{\mathbb{d}}{\mathbb{d}x}\sum_{n=0}^{\infty} f(x) \, \mathbb{d}x = \sum_{n=0}^{\infty} f'(x) \, \mathbb{d}x$$

Question

When taking the derivative of an infinite sum, do you simply differentiate the inner function and take the limit as the iterative variable asymptotically approaches infinity, or is it more complex? In other words, is this a true statement?
$$\frac{\mathbb{d}}{\mathbb{d}x}\sum_{n=0}^{\infty} f(x) \, \mathbb{d}x = \sum_{n=0}^{\infty} f'(x) \, \mathbb{d}x$$

anonymous · Answer

Let's do a proof by example|dw:1388371301481:dw|