Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 9E (Analysis I)

For a polynomial of degree $d$, Taylor expansion about $x_0$ terminates: $P(x_0+h)={\sum }_{n=0}^d{P}^{(n)}(x_0)h^n/n!$.

If $f$ has $n+1$ derivatives near $x_0$, then $f(x)={\sum }_{k=0}^n{f}^{(k)}(x_0)(x-x_0{)}^k/k!+f^{(n+1)}(\xi )(x-x_0{)}^{n+1}/(n+1)!$ for some intermediate $\xi$.

Choose a compact interval around $x_0$ and its constant $C$. For $|h|<1/C$, the Lagrange remainder after order $n$ is at most $(C|h|{)}^{n+1}\to 0$. Thus the taylor series with $a_n=f^{(n)}(x_0)/n!$ equals $f(x_0+h)$.

$e^x$ is real analytic because $e^{x_0+h}=e^{x_0}{\sum }_{n\ge 0}{h}^n/n!$; it is not a polynomial since all its derivatives are nonzero.