Definition: Series

A sequence is defined in the book as a function whose domain in $\mathbf{N}$ (see this). A series is defined as follows:

Let $(b_n)$ be a sequence. A infinite series is a formal expression of the form $$ \begin{align*} \sum_{n=1}^{\infty} b_n = b_1 + b_2 + b_3 + b_4 + ... \end{align*} $$

Here is another definition that I found online:

A Series is a function from the set of infinite sequences to the set of numbers.

I’ve also seen that a series is simply a sum while a sequence is a list.

Definition: Convergence of a Series

When does a given series converge?

Let $(b_n)$ be a sequence and let the infinite series be of the form $$ \begin{align*} \sum_{n=1}^{\infty} b_n = b_1 + b_2 + b_3 + b_4 + ... \end{align*} $$ We define the corresponding sequence of partial sums $(s_m)$ by $$ \begin{align*} s_m = b_1 + b_2 + b_3 + ... + b_m, \end{align*} $$ and say that the series $\sum_{n=1}^{\infty} b_n$ converges to $B$ if the sequence $(s_m)$ converges to $B$. In this case, we write $\sum_{n=1}^{\infty} b_n = B$

What is this even saying? Suppose for a concrete example we’re given the series

$$ \begin{align*} \sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8}. \end{align*} $$

We want to know if this sum converges to something. How do determine that? The idea is to consider instead the sequence of partial sums. Each term in this sequence is defined as $s_m = b_1 + b_2 + b_3 + ... + b_m$. So the first few terms will be:

$$ \begin{align*} s_1 &= \frac{1}{2} \\ s_2 &= \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \\ s_3 &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8} \\ ... \end{align*} $$

And the sequence of partial sum $(s_m)$ is

$$ \begin{align*} \frac{1}{2}, \frac{3}{4}, \frac{7}{8}, ..... \end{align*} $$

The idea is to see if this sequence (the sequence of partial sums $(s_m)$) converges. If we prove that this sequence converges to 1, then we can conclude by the definition above that the infinite sum $\sum_{n=1}^{\infty} \frac{1}{2^n}$ also converges to 1. It’s important to stress here that through analyzing the sequence of partial sums, we’re going to be able to conclude whether the infinite series itself convergences. Putting it in another way,

Let $(b_n)$ be a sequence and let the infinite series be of the form $$ \begin{align*} \sum_{k=1}^{\infty} b_k = B \quad \text{means that} \quad \lim s_n = B \end{align*} $$

(2.7.5) Definition: The Geometric Series

A series is called geometric if it is of the form $$ \begin{align*} \sum_{k=1}^{\infty} ar^k = a + ar + ar^2 + ar^3 + ... \end{align*} $$

For more about its convergence see this.

(2.7.8) Definition: Absolute Convergence

If $\sum_{n=1}^{\infty} |a_n|$ converges, then we say that the original series $\sum_{n=1}^{\infty} a_n$ converges absolutely. If on the other hand, the series $\sum_{n=1}^{\infty} |a_n|$ does not converge, then we say that the original series $\sum_{n=1}^{\infty} a_n$ converges conditionally.

As an example $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges conditionally while $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$ converges absolutely.

(2.7.9) Definition: Rearrangement

Let $\sum_{n=1}^{\infty} a_n$ be a series, A series $\sum_{n=1}^{\infty} b_n$ is called a rearrangement of $\sum_{n=1}^{\infty} a_n$ if there exists a one-to-one, onto function $\mathbf{N} \rightarrow \mathbf{N}$ such that $b_{f(n)} = a_n$ for all $n \in \mathbf{N}$.

(2.7.10) Theorem: Rearrangement

If a series converges absolutely, then any rearrangement of this series converges to the same limit.

Series Definitions