Sequences and Series

A sequence is an ordered infinite list of numbers a1, a2, a3, , while a series is the sum of those terms: n=1^ an.

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: $\sum_{n=1}^{\infty} a_n$ .

Concept	Definition	Example
Sequence	List approaching limit $L$	$\frac{1}{n} \to 0$
Series	Sum of partial sums $S_N$	$\sum_{n=0}^{\infty} \frac{1}{2^n} = 2$

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 $n = 0$ .

A sequence's terms shrinking to a point, and a series' partial sums

Tests and Pitfalls

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

Rule / Test	Condition	Outcome
Term Test	$a_n$ does not go to $0$	Series diverges
Harmonic Series	$\sum \frac{1}{n}$ terms go to $0$	Series diverges

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

Adding Infinitely Many Terms

Tests and Pitfalls

Practise this lesson

Calculus