Sequences and Series

Calculus is the study of the infinite. While derivatives and integrals use infinite processes to find slopes and areas, sequences and series study infinite processes directly. How can you add up an infinite number of things and get a finite answer? This question is at the heart of Zeno’s paradoxes and modern data compression.

Limits of Sequences

A sequence is an infinite list of numbers $a_1, a_2, a_3, \dots$. We are interested in its convergence: does the sequence eventually settle down to a specific value $L$? $\lim_{n \to \infty} a_n = L$

In computer science, every iterative algorithm (like Newton’s method or Gradient Descent) is a sequence. We need to prove that these sequences converge to the correct solution.

Infinite Series: Summing to Infinity

A series is the sum of the terms of a sequence: $\sum_{n=1}^\infty a_n$. A series converges if the sequence of its partial sums $S_n = a_1 + \dots + a_n$ converges to a finite limit.

The Geometric Series

The most famous convergent series is the geometric series: $1 + r + r^2 + r^3 + \dots = \frac{1}{1-r} \quad \text{for } |r| < 1$ This formula is used to calculate the “Present Value” of future cash flows in finance and the “Multiplier Effect” in economics.

Taylor Series: Polynomials for Everything

The ultimate application of series is the Taylor Series. It allows us to represent transcendental functions ($e^x, \sin x, \ln x$) as infinite polynomials. $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$

This is how calculators and computers compute $\sin(1)$. They don’t have a giant lookup table; they use the first few terms of the Taylor series to get as much precision as needed.

import numpy as np
import matplotlib.pyplot as plt

def taylor_exp(x, n):
  """Approximate e^x using first n terms of Taylor series at a=0."""
  approx = 0
  factorial = 1
  for i in range(n):
      if i > 0: factorial *= i
      approx += (x**i) / factorial
  return approx

x_val = 1.0
print(f"Exact e^{x_val}: {np.exp(x_val)}")
for n in [1, 3, 5, 10]:
  print(f"Taylor (n={n}): {taylor_exp(x_val, n)}")

# Visualizing approximation
x = np.linspace(-2, 2, 100)
plt.plot(x, np.exp(x), 'k', label='Exact exp(x)')
plt.plot(x, taylor_exp(x, 3), '--', label='n=3 Approximation')
plt.plot(x, taylor_exp(x, 5), ':', label='n=5 Approximation')
plt.ylim(-1, 8)
plt.legend()
plt.title("Taylor Expansion of e^x")
plt.grid(True)
plt.show()

Power Series and Convergence

A Power Series $\sum c_n x^n$ only converges for $x$ within a certain “Radius of Convergence.” Outside this range, the series explodes to infinity. Knowing this limit is vital for ensuring the stability of electronic filters and signal processing algorithms.

Exercises