Sigma Notation: The Summation Symbol Explained

Every sigma notation expression has four parts: the index variable, a lower limit, an upper limit, and an expression (sometimes called the summand). A typical sum looks like \(\sum_{i=1}^{5} (2i+1)\), which is read as "the sum, as \(i\) goes from 1 to 5, of \(2i+1\)."

• Index variable: usually \(i\), \(j\), \(k\), or \(n\). It changes value with each term.
• Lower limit: the starting value of the index, written below \(\Sigma\).
• Upper limit: the ending value of the index, written above \(\Sigma\).
• Expression: the rule applied to each value of the index, written to the right of \(\Sigma\).

To expand a sigma notation expression, substitute each whole number from the lower limit to the upper limit into the expression, one at a time, then add the results.

Example: Evaluate \(\sum_{i=1}^{5} (2i+1)\).

Substitute \(i = 1, 2, 3, 4, 5\):

\((2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) + (2(5)+1) = 3 + 5 + 7 + 9 + 11 = 35\)

The graph below shows the value of the expression \(2x+1\) at each index from 1 to 5, the same five terms that were added together above.

Example: Evaluate \(\sum_{k=1}^{4} k^2\).

\(1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30\)

A few rules make sums easier to manipulate before you evaluate them.

Constant Factor Rule

A constant multiplied by every term can be factored out of the sum: \(\sum_{i=1}^{n} c \cdot a_i = c \sum_{i=1}^{n} a_i\).

Sum of a Constant

Adding the same constant \(n\) times gives \(\sum_{i=1}^{n} c = c \cdot n\).

Splitting a Sum

A sum of two expressions can be split into two separate sums: \(\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i\).

These properties are especially useful when the expression inside the sigma is built from a known sequence, such as an arithmetic sequence, since you can often simplify before adding a single number.

Sigma notation is really just a shorthand for writing a series. When the terms follow a constant difference, the sum is an arithmetic series; when the terms follow a constant ratio, the sum is a geometric series. If the upper limit is infinity instead of a fixed number, you get an infinite sum, which only has a finite value under specific conditions covered in infinite geometric series.

Recognizing whether the expression inside \(\Sigma\) matches an arithmetic or geometric pattern is often the first step toward choosing the right formula to evaluate the sum quickly, rather than expanding every single term by hand.

• Forgetting that the index changes by 1 each step unless a different step is stated.
• Mixing up the lower and upper limits when substituting values.
• Trying to "cancel" the index variable, when it is only a placeholder used to generate each term.
• Stopping one term short (or adding one extra term) by not including the upper limit itself.

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