Series and Summation Notation

Definition of a Series

A series is the sum of the terms of a sequence. If ${a_{n}}$ is a sequence, then the corresponding series is written as:

S = a_{1} + a_{2} + a_{3} + \dots

More generally, the sum of the first $n$ terms of a sequence is called the partial sum of the series:

S_{n} = a_{1} + a_{2} + \dots + a_{n} = \sum_{k = 1}^{n} a_{k}

A series can either be finite (a sum of a fixed number of terms) or infinite (a sum of infinitely many terms).

Summation Notation (Sigma Notation)

Summation notation (also called sigma notation) is a compact way of writing the sum of the terms in a sequence. It is represented using the Greek capital letter $Σ$ :

\sum_{k = 1}^{n} a_{k}

Where:

$Σ$ represents summation.
$k$ is the index of summation, typically starting at 1.
$n$ is the upper limit, indicating how many terms to sum.
$a_{k}$ represents the $k$ -th term of the sequence.

Example:

The sum of the first $5$ terms of the sequence $a_{n} = n^{2}$ can be written as:

\sum_{n = 1}^{5} n^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} = 1 + 4 + 9 + 16 + 25 = 55

Partial Sums

A partial sum is the sum of the first $n$ terms of a series. The partial sum of the series ${a_{n}}$ is given by:

S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = \sum_{k = 1}^{n} a_{k}

Partial sums are important because they help analyze whether an infinite series converges or diverges.

Example of Partial Sums:

For the sequence $a_{n} = \frac{1}{n}$ , the partial sums for the first 3 terms are:

S_{1} = 1, S_{2} = 1 + \frac{1}{2} = \frac{3}{2}, S_{3} = 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6}

Infinite Series

An infinite series is the sum of infinitely many terms of a sequence:

S = a_{1} + a_{2} + a_{3} + \dots = \sum_{k = 1}^{\infty} a_{k}

The sum of an infinite series is defined as the limit of the sequence of partial sums. If the partial sums approach a finite number $S$ , we say the series converges to $S$ :

S = lim_{n \to \infty} S_{n}

If the partial sums do not approach a finite limit, we say the series diverges.

Convergence and Divergence of Series

Convergence:

An infinite series $\sum_{n = 1}^{\infty} a_{n}$ converges if the sequence of partial sums ${S_{n}}$ has a finite limit as $n \to \infty$ . That is:

lim_{n \to \infty} S_{n} = L

where $L$ is a real number. The series then sums to $L$ .

Example of a Convergent Series:

Consider the geometric series:

S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots

This series converges to:

S = \frac{1}{1 - \frac{1}{2}} = 2

Divergence:

An infinite series $\sum_{n = 1}^{\infty} a_{n}$ diverges if the partial sums do not approach a finite limit. This may happen if the partial sums grow without bound, or if they oscillate between values without settling on a single number.

Example of a Divergent Series:

The harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ is an example of a divergent series:

S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots

Although the terms get smaller, the partial sums increase without bound, meaning the series diverges.

Tests for Convergence:

There are several tests to determine if a series converges or diverges, including:

nth-Term Test for Divergence: If $lim_{n \to \infty} a_{n} \neq 0$ , then the series diverges.
Geometric Series Test: A geometric series $\sum_{n = 0}^{\infty} a r^{n}$ converges if $| r | < 1$ .
Integral Test: A series $\sum a_{n}$ can be compared to an integral $\int f (x) d x$ to test for convergence.
Comparison Test: Compare the series $\sum a_{n}$ with another known convergent or divergent series to determine the behavior.