Sequences and Series
Sequences and series provide the machinery for infinite sums and approximations. Convergence tests determine when infinite processes are well-defined.
Key Points
- A series converges if its sequence of partial sums converges.
- Absolute convergence implies convergence and allows rearrangement.
- Taylor series approximate smooth functions by polynomials.
Formulas
Taylor series
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
Geometric series
$$\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}, \quad |r| < 1$$
Code Example
import math
def exp_taylor(x, n=10):
return sum(x**k / math.factorial(k) for k in range(n))
print(exp_taylor(1.0)) # ~ e