In this chapter we examine sequences of sums. For a sequence $(a_n)_{n\in\mathbb N}$ we consider the sequence defined by $s_n = a_1 + a_2 + \dots + a_n$ , i.e.

$$\begin{aligned} s_1 &= a_1 \\ s_2 &= a_1 + a_2\\ s_3 &= a_1 + a_2 + a_3\text,\end{aligned}$$ and so on. The sequence $(s_n)_{n\in\mathbb N}$ is called the seqence of partial sums of $(a_n)_{n\in\mathbb N}$ .

The formula $s_n = a_1 + a_2 + \dots + a_n$ is commonly abbreviated with the $\Sigma$ symbol in the following way:

$$s_n = \sum_{k=1}^n a_k\text.$$ The Greek letter $\Sigma$ is also called the summation symbol, and it expresses that the terms on the inside (the $a_k$ ) have to be added.

Under the summation symbol we specify where we start to sum and atop of the summation symbol we specify the upper bound for the sum. For example,

$$\sum_{k=3}^5 \frac{1}{k} = \frac 13 + \frac 14 + \frac 15\text.$$ If we use the summation symbol inside a sentence, the indices will be moved as follows: $\textstyle \sum_{k=3}^5 \frac{1}{k}$ .

For some sequences $(a_n)_{n\in\mathbb N}$ the sequence $(s_n)_{n\in\mathbb N}$ of partial sums converges. In this chapter we will discuss various methods that can help in determining whether $(s_n)_{n\in\mathbb N}$ converges, diverges or neither of these.