Convergence and Divergence Convergence and Divergence
Introduction
A sequence is a list of numbers arranged in a specific order.
1, 2, 3, 4, 5, \ldots
A series is the sum of the terms of a sequence.
1 + 2 + 3 + 4 + 5 + \ldots
In this module, we will be working with infinite series, which are sums of infinitely many terms.
Here is the notation for a series:
\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots
In the notation above:
\sum
The symbol for summation.
1
The starting number (at the bottom).
\infty
The ending number (at the top).
a_n
The general term. Here is the notation for summing the first 100 positive integers, squared:
\sum_{n=1}^{100} n^2 = 1^2 + 2^2 + 3^2 + \cdots + 100^2
Convergence and Divergence
The sum of a convergent series approaches a finite number.
The sum of a divergent series does not.
In other words, if the following limit exists:
\lim_{n \to \infty} \sum_{i=1}^{n} a_i = L
Then the series is convergent.
Otherwise, it is divergent.
Note, for the sake of notation, that:
\lim_{n \to \infty} \sum_{i=1}^{n} a_i=\sum_{n=1}^{\infty} a_n
Examples
Example 1: Convergent Series
\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots=2
Below is not the standard method of proving convergence, but it will help in visualizing it:
The left box has an area of
1
. The next biggest box has an area of
1/2
. The pattern continues, and the sum of all the areas is equal to the area of the sketch:
2
.
Example 2: Divergent Infinite Series \sum_{n=0}^{\infty} (2)^n = 1 + 2 + 4 + 8 + \cdots
Since the terms are all positive and approaching infinity, their sum also approaches infinity.
Therefore, the series diverges.
Example 3: Divergent Oscillating Series \sum_{n=0}^{\infty} (-1)^n = (-1) + 1 + (-1) + \cdots
Since the sequence is oscillating between -1 and 1, the sum is oscillating between -1 and 0. The series does not approach a finite number.
Therefore, the series diverges.