Infinite Series of Real and Complex Numbers

Recall that a sequence of numbers $(a_k)_{k=1}^{\infty} = (a_1, a_2, ..., a_k, ...)$ is simply an infinite ordered list of numbers. Suppose instead that we consider the sum of all of the terms in such a sequence. What we end up getting is an infinite sum of terms in a sequence which we define as a numerical series.

Definition: Let $(a_n)_{n=1}^{\infty} = (a_1, a_2, ..., a_n, ...)$ be an infinite sequence of real/complex numbers. The corresponding Infinite Series is $\displaystyle{\sum_{n=1}^{\infty} a_n = a_1 + a_2 + ... + a_n + ...}$ and is the sum of all terms in the corresponding Sequence of Terms $(a_n)_{n=1}^{\infty}$.

For example, consider the sequence $\left ( \frac{1}{n} \right )_{n=1}^{\infty}$. The corresponding series is:

(1)

\begin{align} \quad \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... \end{align}

We will see later the series above actually sums to $\infty$.

Another important sequence in a series is the sequence of partial sums which we define below.

Definition: If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a series then the corresponding Sequence of Partial Sums denoted $(s_n)_{n=1}^{\infty}$ is the sequence with the general term $\displaystyle{s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + ... + a_n}$. In other words, the term $s_n$ denotes the finite $n^{\mathrm{th}}$ sum of the corresponding infinite series.

For example, the first few terms of the sequence of partial sums for the series $\sum_{k=1}^{\infty} \frac{1}{k}$ are:

(2)

\begin{align} \quad s_1 &= 1 ,\\ \quad s_2 &= 1 + \frac{1}{2} = \frac{3}{2}, \\ \quad s_3 &= 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6}, \\ \quad s_4 &= 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{25}{12} \\ \end{align}

Notice that:

(3)

\begin{align} \quad s_1 &= a_1 \\ \quad s_2 &= a_2 + s_1 \\ \quad s_3 &= a_3 + s_2 \\ \end{align}

And in general, $s_n = a_n + s_{n-1}$. So the $n^{\mathrm{th}}$ term of the sequence of partial sums can be obtained by adding the $n^{\mathrm{th}}$ term of the sequence of terms to the $(n-1)^{\mathrm{th}}$ term of the sequence of partial sums.