How to write a sum in sigma notation

When a sum has several terms, sigma notation gives us a shorter way to write the same addition. We’ll number the terms, find an expression that produces each one, and choose the first and last index values. Let’s start with a simple pattern, then work through a sum with fractions.

Match the term number to the term

Start by labeling the first term 1, the second term 2, and so on. We want an expression that gives the correct term when we substitute its number.

For example, consider the sum

3 + 6 + 9 + 12

The first term is $3 (1)$ , the second is $3 (2)$ , the third is $3 (3)$ , and the fourth is $3 (4)$ . The general term is $3 i$ , and the index runs from 1 to 4, so

3 + 6 + 9 + 12 = \sum_{i = 1}^{4} 3 i

Example: Write a fractional sum in sigma notation

Rewrite the following sum using summation notation, with x nonzero.

\begin{array}{l} \frac{3 x + 1}{7 x} + \frac{6 x + 2}{14 x^{2}} \\ + \frac{9 x + 3}{21 x^{3}} + \frac{12 x + 4}{28 x^{4}} \\ + \frac{15 x + 5}{35 x^{5}} + \frac{18 x + 6}{42 x^{6}} \end{array}

Find the pattern in the numerators

Number the terms with $n = 1, 2, 3, 4, 5, 6$ . The coefficient of x in each numerator is three times the term number, and the added number is the term number itself.

3 (1) x + 1, 3 (2) x + 2, 3 (3) x + 3

That gives the general numerator $3 n x + n$ . For example, substituting $n = 4$ gives $12 x + 4$ , which matches the fourth term.

Find the pattern in the denominators

The numerical coefficients in the denominators are 7, 14, 21, 28, 35, and 42. Each one is seven times the term number. The exponent on x also matches the term number.

The general denominator is therefore $7 n x^{n}$ . When $n = 4$ , it gives $28 x^{4}$ . The complete general term is

\frac{3 n x + n}{7 n x^{n}}

Choose the bounds and write the sum

There are six terms, starting with term 1 and ending with term 6. Put those bounds below and above sigma.

\sum_{n = 1}^{6} \frac{3 n x + n}{7 n x^{n}}

We can check the pattern by substituting $n = 1$ and $n = 6$ . Those values give the first and last fractions in the original sum, and the values between them give the other four terms.

Can a different sigma expression represent the same sum?

Yes. The index letter is a label, so changing n to i does not change the sum if the same letter is used throughout the expression. We can also start counting from a different index value if we adjust both the general term and the bounds to produce the same terms.

For the reverse process, see how to expand summation notation.

Learn mathKrista KingApril 5, 2016summation notation, sum, online math course, magic math