Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.

[latex]3+7+11+15+19+\cdots[/latex]

The [latex]n\text{th }[/latex] partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation

[latex]\begin{align}&{S}_{n}\text{ represents the partial sum.} \\ &{S}_{1}=3 \\ &{S}_{2}=3+7=10 \\ &{S}_{3}=3+7+11=21 \\ &{S}_{4}=3+7+11+15=36\end{align}[/latex]

Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, [latex]\Sigma[/latex], to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.

If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{k}=2k[/latex] for [latex]k=1[/latex] through [latex]k=5[/latex]. We can begin by substituting the terms for [latex]k[/latex] and listing out the terms of this series.

[latex]\begin{align} &{a}_{1}=2\left(1\right)=2 \\ &{a}_{2}=2\left(2\right)=4 \\ &{a}_{3}=2\left(3\right)=6 \\ &{a}_{4}=2\left(4\right)=8 \\ &{a}_{5}=2\left(5\right)=10 \end{align}[/latex]

We can find the sum of the series by adding the terms:

[latex]\sum\limits _{k=1}^{5}2k=2+4+6+8+10=30[/latex]

Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference, [latex]d[/latex]. The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first [latex]n[/latex] terms of an arithmetic series as:

[latex]{S}_{n}={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+…+\left({a}_{n}-d\right)+{a}_{n}[/latex].

We can also reverse the order of the terms and write the sum as

[latex]{S}_{n}={a}_{n}+\left({a}_{n}-d\right)+\left({a}_{n}-2d\right)+…+\left({a}_{1}+d\right)+{a}_{1}[/latex].

If we add these two expressions for the sum of the first [latex]n[/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[/latex] terms of any arithmetic series.

[latex]\begin{align}{S}_{n}&={a}_{1}+\left({a}_{1}+d\right)+\left({a}_{1}+2d\right)+…+\left({a}_{n}-d\right)+{a}_{n} \\ +{S}_{n}&={a}_{n}+\left({a}_{n}-d\right)+\left({a}_{n}-2d\right)+…+\left({a}_{1}+d\right)+{a}_{1} \\ \hline 2{S}_{n}&=\left({a}_{1}+{a}_{n}\right)+\left({a}_{1}+{a}_{n}\right)+…+\left({a}_{1}+{a}_{n}\right) \end{align}[/latex]

Because there are [latex]n[/latex] terms in the series, we can simplify this sum to

[latex]2{S}_{n}=n\left({a}_{1}+{a}_{n}\right)[/latex].

We divide by 2 to find the formula for the sum of the first [latex]n[/latex] terms of an arithmetic series.

[latex]{S}_{n}=\dfrac{n\left({a}_{1}+{a}_{n}\right)}{2}[/latex]

This is generally referred to as the Partial Sum of the series.