Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 11F (Analysis I)

The Taylor theorem with Lagrange remainder states that if $f$ is $n$ times differentiable between $a$ and $x$, then some $\xi$ between $a$ and $x$ satisfies

Let $T_{n-1}$ denote the displayed polynomial and choose $K$ so thatsatisfies $F(x)=0$. By construction,Starting with the two zeros $a,x$ and applying Rolle theorem repeatedly, there is a $\xi$ between them with $F^{(n)}(\xi )=0$. Sincewe have $K=f^{(n)}(\xi )/n!$, proving the theorem.

Repeatedly differentiating $f^{\prime \prime }=-f$ and using $f(0)=1$, $f^{\prime }(0)=0$ givesAlso,so $f^2+(f^{\prime }{)}^2=1$. Every derivative is one of $\pm f,\pm f^{\prime }$, and hence has absolute value at most one.

Taylor's theorem at zero through degree $2N+1$ giveswherefor each fixed $x$. Therefore