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