Series Solution of Nonlinear ODE I am trying to solve for the series solution of the following nonlinear ODE: $$y'+y^2=0$$ I am stuck in the following step: $$\sum_{n=1}^\infty a_nnx^{n-1}+\sum_{n=0}^\infty \left ( \sum_{k=0}^n a_ka_{n-k} \right )x^n=0$$ Is it possible to solve the DE this way?

It might be more efficient to attack the problem with force if the general pattern is not readily seen.
Let
\begin{align}
y(x) = \sum_{n=0}^{\infty} a_{n} \, x^{n} 
\end{align}
for which the equation $y' + y^{2} = 0$ leads to
\begin{align}
a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \cdots &= - (a_{0} + a_{1} x + a_{2} x^{2} + \cdots )^{2} \\
&= - [ a_{0}^{2} + (a_{0}a_{1} + a_{1} a_{0}) x + \cdots ]. 
\end{align}
Equating the coefficients leads to
\begin{align}
a_{1} &= - a_{0}^{2} \\
2 a_{2} &= - (a_{0}a_{1} + a_{1} a_{0}) 
\end{align}
and so on. What is then determined is the general form of the coefficients being
\begin{align}
(n+1) \, a_{n} = - \sum_{k=0}^{n} a_{k} \, a_{n-k}
\end{align}
