I have a question about derivatives of summations while working with ODEs. From what I know (and have seen up until now), to take the derivative of a summation, such as a power series, the index of the sum has to increase by 1 to account for the loss of the constant in the original sum:
$(\sum_{n=0}^{\infty} c_n x^n)'=\sum_{n=1}^{\infty} n c_n x^{n-1}$
However, while working on the Frobenius' method for solving ODEs about singular points, I have found that the derivatives of indexes do not change, such that, if $y=\sum_{n=0}^{\infty} c_n x^{n+r}$, then
$y'=\sum_{n=0}^{\infty} (n+r) c_n x^{n+r-1}$.
And similarly for $y''$. I have no idea why this "exception" is the case. Could anyone help with this? Many thanks!
As eyeballfrog's comment suggests, your first sum is the exception, whereas the second is the rule.
Consider $$y(x)=\sum_{n=0}^{\infty} c_n x^{n+r},$$ then $$\frac{dy}{dx}=\sum_{n=0}^{\infty} (n+r) c_n x^{n+r}.$$
For $r\neq 0$, that sum stays as it is whereas for $r=0$ we have
$$\frac{dy}{dx}=\sum_{n=0}^{\infty} (n+r) c_n x^{n+r}=\sum_{n=0}^{\infty} (n) c_n x^{n}=\sum_{n=1}^{\infty} n c_n x^{n},$$
because the $n=0$ term equates to $0\times c_0 \times x^0 =0$.
Note that when computing the derivative of the formal power series, we interchange summation and differentiation, which is possible only under certain conditions.