Taylor Series Error

Taylor Series Error

The error of a degree-$n$ Taylor approximation is:

$E_n(x) = |f(x) - P_n(x)|$

Taylor's Remainder Theorem

There exists some $c$ between $a$ and $x$ such that:

E_n(x) = rac{|f^{(n+1)}(c)|}{(n+1)!} cdot |x-a|^{n+1}

This bounds the error by the next term of the series (at the worst $c$).

Intuition

• The degree-$n$ polynomial matches $f$ through all derivatives up to order $n$ at $a$
• The first "mismatch" comes from the $(n+1)$-th derivative
• The further $x$ is from $a$, the larger the error (raised to the $n+1$ power)

Example: $x^3$ with $P_2$

For $f(x) = x^3$, the Taylor polynomial at $a=0$ with $n=2$:

P_2(x) = f(0) + f'(0)x + rac{f''(0)}{2!} x^2 = 0 + 0 + 0 = 0

At $x=2$: error $= |8 - 0| =$ 8.

The bound from the theorem: rac{|f'''(c)|}{3!} cdot |2|^3 = rac{6}{6} cdot 8 = 8. Tight!

Convergence Check

For polynomial $f(x) = x^k$, a Taylor polynomial of degree $k$ at $a=0$ has zero error (it's exact).

Your Task

Implement double taylor_error(double (*f)(double), double a, double x, int n, double h) that returns $|f(x) - P_n(x)|$.

Helper functions from the Taylor polynomial lesson are provided.