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Pre-Calculus Sequences and series (advanced)

Sigma notation (advanced)

20 practice questions 0 video lessons Theory + worked examples

Sigma Notation

Texas Precalculus (TEKS) • Standard P.5(A), P.5(D) • Sequences & Series

Sigma Notation is a topic in Sequences & Series in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.5(A), P.5(D), which requires students to evaluate finite sums and represent series using sigma notation.

Sigma notation \(\displaystyle\sum_{k=1}^{n}a_k\) writes a sum compactly, with the index running from a lower to an upper limit.

Texas Precalculus (TEKS) › Sequences & Series › Sigma Notation  —  Standard P.5(A), P.5(D)

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Theory

Sigma notation writes a sum compactly using the Greek letter \(\Sigma\):
\[\sum_{k=1}^{n}a_k=a_1+a_2+\cdots+a_n.\]

The index \(k\) starts at the lower limit and runs to the upper limit, and \(a_k\) is the rule for each term.

Sums are linear: you can factor out constants and split sums apart.

Handy formulas: \(\displaystyle\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\) and \(\displaystyle\sum_{k=1}^{n}c=cn\).
Sigma notation Sigma notation compactly writes a sum: the index starts at the bottom value and runs to the top value. ∑nk=1aₖ= a₁+a₂+…+aₙupper limitlower limit & index
Reading sigma notation: index, limits, and term rule.
Sum properties Sum properties Sum properties ∑ c·aₖ = c ∑ aₖ ∑(aₖ+bₖ) = ∑aₖ + ∑bₖ ∑ₙ c = c·n
The linearity and constant properties.

Properties and two standard sums:

\[\sum c\,a_k=c\sum a_k,\qquad \sum(a_k+b_k)=\sum a_k+\sum b_k\]
\[\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2},\qquad \sum_{k=1}^{n}c=cn\]
constants factor out of a sum; sums split over addition; the sum of 1 to n is n times n plus 1 over 2
Change the index carefully if you shift the limits — the term rule must move with it.

How to evaluate a sum

  1. Read the index, its limits, and the term rule.
  2. Expand for a few terms, or apply a formula for many.
  3. Use properties to factor constants and split sums.
  4. Add or apply \(\dfrac{n(n+1)}{2}\), \(cn\), etc.
Example 1 — Expand a sum
Write out \(\displaystyle\sum_{k=1}^{4}(2k+1)\) and evaluate.
Solution

Substitute \(k=1,2,3,4\) and add.

\(=\)3+5+7+9
\(=\)24
the sum is 24
Example 2 — A constant sum
Evaluate \(\displaystyle\sum_{k=1}^{10}7\).
Solution

Adding a constant \(n\) times gives \(cn\).

\(=\)7\cdot 10=70
the sum is 70
Example 3 — Use a known formula
Evaluate \(\displaystyle\sum_{k=1}^{100}k\).
Solution

Use \(\displaystyle\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\).

\(=\)\dfrac{100\cdot 101}{2}
\(=\)5050
the sum of 1 to 100 is 5050
Example 4 — Split with properties
Evaluate \(\displaystyle\sum_{k=1}^{5}(3k)\).
Solution

Factor out the constant, then sum.

\(=\)3\sum_{k=1}^{5}k
\(=\)3\cdot 15=45
the sum is 45

Common pitfalls

Include both limits. \(\sum_{k=1}^{4}\) has four terms, \(k=1,2,3,4\).
Constants still get summed. \(\sum_{k=1}^{n}c=cn\), not \(c\).
Factor constants, don't drop them. \(\sum 3k=3\sum k\).

Frequently asked questions

What is sigma notation?

A compact way to write a sum, \(\sum_{k=1}^{n}a_k\), where the index \(k\) runs from the lower to the upper limit.

How do you expand a sigma sum?

Substitute each index value from the lower to the upper limit and add the results.

Can you factor a constant out of a sum?

Yes: \(\sum c\,a_k=c\sum a_k\). Sums are linear.

What is the sum of the first n whole numbers?

\(\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\).

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