Sequences, Series and Geometric Series

Sequences

A sequence is an infinite list of numbers $a_1, a_2, a_3, \ldots$, usually given by a formula for $a_n$. It converges to $L$ if its terms get as close to $L$ as we like and stay there: $\lim_{n \to \infty} a_n = L$. Otherwise it diverges.

• $a_n = \tfrac{1}{n}$: the terms $1, \tfrac{1}{2}, \tfrac{1}{3}, \ldots$ converge to $0$.
• $a_n = (-1)^n$: the terms $-1, 1, -1, 1, \ldots$ never settle, so the sequence diverges.
• $a_n = \tfrac{3n + 1}{2n - 5}$: divide top and bottom by $n$, and the limit is $\tfrac{3}{2}$, the same as for a rational function at infinity.

Series and partial sums

A series adds up the terms of a sequence: $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$. An infinite sum cannot be done one addition at a time, so its value is defined through the partial sums

$$S_n = a_1 + a_2 + \cdots + a_n$$

If the partial sums converge to $S$, the series converges and its sum is $S$. Otherwise the series diverges. A series is a sequence in disguise: the sequence of its partial sums.

• the sum, 1

Geometric series

A geometric series multiplies by the same ratio $r$ at every step: $a + ar + ar^2 + \cdots$. Its partial sums have a closed form. Subtracting $rS_n$ from $S_n$ cancels all but two terms, so

$$S_n = a\,\frac{1 - r^n}{1 - r} \qquad (r \ne 1)$$

If $\lvert r \rvert < 1$, then $r^n \to 0$ and the series converges:

$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \qquad \text{for } \lvert r \rvert < 1$$

If $\lvert r \rvert \ge 1$ and $a \ne 0$, the terms do not shrink to $0$ and the series diverges.

Why an infinite sum can be finite

Cut a square of area $1$ in half, then cut the remaining half in half, and keep going. The pieces have areas $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots$, and together they fill the square exactly: the part left over after $n$ cuts is $\left(\tfrac{1}{2}\right)^n$, which shrinks to nothing.

An infinite series has a sum exactly when its partial sums settle down; a geometric series with $\lvert r \rvert < 1$ settles because each term closes a fixed fraction of the remaining gap.

The nth-term test

If a series converges, its terms must shrink to $0$: each term is the difference $S_n - S_{n-1}$ of two partial sums approaching the same limit. Turned around, this gives a test for divergence.

The nth-term test

If $\lim_{n \to \infty} a_n \ne 0$, or the limit does not exist, then $\sum a_n$ diverges.

The test never proves convergence. The terms of $1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} + \cdots$ shrink to $0$, yet the next lesson shows that this series diverges.

Worked examples

Common mistakes

Practice problems