Tutorial 12: Taylor & Maclaurin Series

Tutorial problems covering power series expansions and approximations.

Sections

Maclaurin Series (Problems 1-4)

  • Finding Maclaurin series from definition
  • Using known series
  • Radius of convergence

Taylor Series (Problems 5-8)

  • Series about specific points
  • Taylor polynomials
  • Remainder estimation

Applications (Problems 9-12)

  • Function approximation
  • Numerical calculations
  • Limit evaluation using series

Standard Maclaurin Series

Function Series
$e^x$ $\sum_{n=0}^{\infty} \frac{x^n}{n!}$
$\sin x$ $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
$\cos x$ $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$
$\ln(1+x)$ $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$
$(1+x)^n$ $\sum_{r=0}^{\infty} \binom{n}{r} x^r$

Taylor Series Formula

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Applications

  • Approximating function values
  • Evaluating limits
  • Integration of non-elementary functions
  • Solving differential equations

Links