Taylor and Power Series

A Taylor series is an infinite sum of terms that are expressed in terms of the function’s derivatives at a single point.

Taylor’s Formula

The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable at a real or complex number $a$ is the power series: $f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots$

import numpy as np

def sin_taylor(x, n_terms):
  res = 0
  for n in range(n_terms):
      sign = (-1)**n
      term = (x**(2*n + 1)) / np.math.factorial(2*n + 1)
      res += sign * term
  return res

x_val = np.pi / 4 # 45 degrees
print(f"sin(pi/4) exact: {np.sin(x_val):.6f}")
print(f"Taylor (1 term):  {sin_taylor(x_val, 1):.6f}")
print(f"Taylor (3 terms): {sin_taylor(x_val, 3):.6f}")

Radius of Convergence

A power series $\sum a_n (x-a)^n$ has a radius of convergence $R$ such that the series converges absolutely for $|x-a| < R$ and diverges for $|x-a| > R$.

Remainder Term: Taylor’s Theorem

Taylor’s Theorem gives an estimate of the error when a function is approximated by its $n$-th degree Taylor polynomial. The Lagrange form of the remainder is: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{(n+1)}$ for some $c$ between $a$ and $x$.

def exp_error_estimate(x, n):
  import math
  # Error for exp(x) is x^(n+1)/(n+1)! * exp(c)
  # where c is in [0, x]. Max error at c=x.
  return (abs(x)**(n+1) / math.factorial(n+1)) * math.exp(abs(x))

print(f"Max error for exp(1) with 5 terms: {exp_error_estimate(1, 5):.6f}")