In the Taylor series

$\displaystyle u(x+h)=u(x)+hu'(x)+\frac{1}{2}h^2 u''(x)+\frac{1}{6}h^3 u'''(x)+...$

find the number a in the next term $\displaystyle a h^4 u''''(x)$ by testing $\displaystyle u(x)=x^4$ at x=0

OK, so, I take the 4th derivative of u(x): u''''(x)=24 and u''''(4)=24

Now I have $\displaystyle a h^4 24$ as my next term, but how does that help me find the constant a? I'm confused.

Ah, wait a minute, I see it.

$\displaystyle u(x+h)=(x+h)^4=h^4+4 h^3 x+6 h^2 x^2+4 h x^3+x^4$

The $\displaystyle h^4$ term has a coefficient of 1. So $\displaystyle 1=a 24$ and $\displaystyle a=\frac{1}{24}$