Chapter 11.2: Series

Welcome to the fascinating world of series! In this section, we'll explore what happens when we try to add up the terms of an infinite sequence. Get ready to understand when these sums converge to a finite value and when they diverge!

What is a Series?

When we express a number as an infinite decimal, we're actually representing it as an infinite sum. For example, consider $\pi = 3.14159...$. This can be written as:

$$\pi = 3 + \frac{1}{10} + \frac{4}{10^2} + \frac{1}{10^3} + \frac{5}{10^4} + ...$$

In general, if we have an infinite sequence ${a_n}_{n=1}^{\infty}$, adding all the terms together gives us an infinite series:

$$a_1 + a_2 + a_3 + ... + a_n + ...$$

This series is denoted by the symbol:

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

Partial Sums

To determine if a series has a sum, we look at its partial sums. The partial sums are defined as follows:

  • $s_1 = a_1$
  • $s_2 = a_1 + a_2$
  • $s_3 = a_1 + a_2 + a_3$
  • In general, $s_n = a_1 + a_2 + ... + a_n = \sum_{i=1}^{n} a_i$

The sequence of partial sums, ${s_n}$, may or may not have a limit.

Convergence and Divergence

Definition: Given a series $\sum_{n=1}^{\infty} a_n$, if $\lim_{n \to \infty} s_n = s$ (where $s$ is a finite number), then the series is called convergent, and we write:

$$\sum_{n=1}^{\infty} a_n = s$$

The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.

In other words, the sum of a series is the limit of the sequence of its partial sums.

Example: A Geometric Series

A very important example of an infinite series is the geometric series:

$$a + ar + ar^2 + ar^3 + ... = \sum_{n=1}^{\infty} ar^{n-1}, \quad a \neq 0$$

Each term is obtained by multiplying the previous one by the common ratio $r$. If $|r| < 1$, the geometric series converges, and its sum is:

$$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}, \quad |r| < 1$$

If $|r| \geq 1$, the geometric series diverges.

Test for Divergence

An important test to quickly check if a series diverges is the Test for Divergence:

If $\lim_{n \to \infty} a_n$ does not exist or if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ is divergent.

Keep practicing, and you'll master the art of determining whether series converge or diverge. Good luck, and have fun!