How to Write a Series in Sigma Notation

Sigma notation is a compact way to write a sum. The Greek letter sigma, $\sum$, tells you to add terms generated by a formula as an index runs from a starting value to an ending value. To convert a written series into sigma notation, identify the rule for its terms and choose bounds that include precisely the terms you want.

A series written with an ellipsis does not always have a unique rule. The examples below use the natural patterns indicated by their terms; when a problem gives a rule, use that rule rather than guessing from a few terms.

The parts of sigma notation

A finite sum has the form

$$\sum_{k=m}^{n} a_k.$$

Here, $k$ is the index, $m$ is its starting value, $n$ is its ending value, and $a_k$ is the general term. Expand the notation by substituting each integer value of $k$ from $m$ through $n$ and adding the resulting terms:

$$\sum_{k=m}^{n} a_k = a_m + a_{m+1} + \cdots + a_n.$$

The bounds are inclusive. Thus, $\sum_{k=1}^{5} a_k$ contains five terms, not four. The letter used for the index is arbitrary: $\sum_{k=1}^{5} k$ and $\sum_{j=1}^{5} j$ mean the same thing.

For an infinite series, the upper bound is infinity:

$$\sum_{k=m}^{\infty} a_k.$$

This notation represents an infinite sum. Whether it has a finite value depends on whether the series converges; the sigma notation alone does not guarantee convergence.

A step-by-step method

1. Count or characterize the terms. Decide whether the sum is finite or infinite, and identify the first and last terms when it is finite.
2. Choose an index. Use a convenient integer variable such as $k$ or $n$. The choice does not affect the sum.
3. Find the general term. Write a formula that produces the term associated with each index.
4. Choose the bounds. Make the first index produce the first term and the last index produce the final term.
5. Check by expanding. Substitute the first few index values. This catches most errors with starting indices, signs, and exponents.

Worked example: consecutive even numbers

Write

$$2 + 4 + 6 + 8 + 10$$

in sigma notation. The terms are twice the counting numbers: the first term is $2(1)$, the second is $2(2)$, and the fifth is $2(5)$. So the general term is $2k$, with $k$ running from 1 to 5:

$$2 + 4 + 6 + 8 + 10 = \sum_{k=1}^{5} 2k.$$

Check the endpoints: $2(1)=2$ and $2(5)=10$. The terms in between are $2(2)=4$, $2(3)=6$, and $2(4)=8$, so the expression matches the whole series.

Worked example: alternating reciprocals

Consider the finite series

$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5}.$$

The denominator of each term is its position, so the magnitude is $1/k$. The signs alternate, starting positive. The factor $(-1)^{k+1}$ is positive when $k=1$, negative when $k=2$, and continues alternating. Therefore,

$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} = \sum_{k=1}^{5} \frac{(-1)^{k+1}}{k}.$$

If the pattern continues forever, the corresponding notation is $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}$. The finite and infinite sums are related, but they are not the same expression: one has five terms, while the other has infinitely many.

Starting at a different index

You do not have to start the index at 1. For example,

$$5 + 6 + 7 + 8 = \sum_{k=5}^{8} k.$$

You could also start at 1 by changing the formula: $\sum_{k=1}^{4} (k+4)$. Both expressions generate the same terms. Pick the version that makes the term rule and bounds easiest to see.

Common mistakes

• Using bounds that generate the wrong number of terms. From $k=1$ through $k=5$ there are five integer values. Check the first and last substitutions.
• Getting the sign pattern backward. Test the exponent in an alternating factor at the first index. For example, $(-1)^{k+1}$ starts positive at $k=1$, while $(-1)^k$ starts negative.
• Confusing a term with a sum. The formula $a_k$ describes one term; $\sum a_k$ means to add the terms over the specified index values.
• Assuming ellipses determine a unique rule. A few terms can fit many patterns. Use any stated formula or context, and mention your assumption if the pattern is not specified.
• Treating infinity as an ordinary endpoint. An infinite series has infinitely many terms, and its convergence must be checked separately.

When in doubt, expand your proposed sigma notation for its first several index values. If the resulting terms match the series and the bounds cover the intended range, the notation is correct.