In an example in my textbook, it is mentioned that the sequence generated by:
${f(x)= e^x + x^2}$
is: 1,1,3,1,1,1,1,...
why is it that when $x^2$ is added to $\sum_{i=0}^{\infty} x^i/i!$ we would get 3 as the coefficient of the third term in this sequence?
To elaborate on André's comment: the sequence $(1,1,3,1,1,\ldots)$ is obtained by reading the coefficients of $x^k/k!$, not the coefficients of $x^k$. The coefficient of $x^2/2!$ is 3: \begin{eqnarray*} e^x + x^2 &=& x^2 + \sum_{k=0}^\infty \frac{x^k}{k!}\\ &=& x^0 + x^1 + 3\frac{x^2}{2!} + \sum_{k=3}^\infty \frac{x^k}{k!}. \end{eqnarray*}