Sequences and Series

Adding Infinitely Many Terms

A sequence is an ordered infinite list of numbers $a_1, a_2, a_3, \dots$, while a series is the sum of those terms: $\sum_{n=1}^{\infty} a_n$.

Convergence asks if adding infinite terms yields a finite total. A geometric series $\sum r^n$ converges when $|r| < 1$, summing to $\frac{1}{1 - r}$ for $n = 0$.

Tests and Pitfalls

Series are vital in computing: they define functions, analyze algorithm running times, and power numerical methods.

Common pitfall: Assuming terms shrinking to zero guarantees convergence. That is false. Shrinking to zero is necessary but not sufficient.