http://mathworld.wolfram.com/FrobeniusMethod.html
Was reading this and was wondering why the $n$ does not increase while computing the derivatives of $y$ in the frobenius method. In the first derivative for example, shouldn't the sum be from $n=1$ to $\infty$?
The term corresponding to $n=0$ in the expression for $y$ is $a_0 x^{k}$, and the derivative of this term is $a_0 k x^{k-1}$, which is the term corresponding to $n=0$ in the expression for $y'$. So we really need to start summing at $n=0$ to include all the terms that should be present in $y'$.
Unless $k=0$, in which case the zeroth term $a_0 k x^k$ simply equals zero, and then we may as well start the summation on $n=1$ in $y'$ (or we may still start on $n=0$, it wouldn't make any difference). This case might be familiar to you from earlier experience with looking for ordinary power series solutions to linear ODEs (like in equations (2) and (3) on the page you linked to), without that extra factor $x^k$ in front, so maybe that's why you were expecting $y'=\sum_1^\infty$?
