Understand and use sigma notation for sums of series.

The idea

Sigma notation compactly writes a sum. In Σ from r = 1 to n of uᵣ, the letter r is the index, 1 is the lower limit, n is the upper limit, and uᵣ is the general term. To evaluate, substitute each integer index from the lower to the upper limit and add the results. For example, Σ from r = 1 to 4 of (2r + 1) gives 3 + 5 + 7 + 9 = 24. Sigma notation works with arithmetic and geometric series, and standard results such as Σr = n(n + 1)/2 and Σr² = n(n + 1)(2n + 1)/6 allow sums to be found without listing every term. You must also be able to rewrite an expanded sum in sigma form and adjust limits when the index starts at a value other than 1.

What earns the marks

• Interprets each part of sigma notation correctly: index, lower limit, upper limit and general term.
• Evaluates a finite sum by substituting consecutive integer values of the index and adding the terms.
• Rewrites an expanded series using sigma notation with a correct general term and limits.
• Applies standard summation results such as Σr = n(n + 1)/2 and Σr² = n(n + 1)(2n + 1)/6.
• Handles sums whose index starts at a value other than 1 by adjusting the general term or subtracting a partial sum.

Common mistakes and how to put them right

• Substituting the upper limit only instead of every integer from the lower to the upper limit.

  Put it right: List each index value in turn.

• Confusing the index letter with a variable in the general term and treating it as fixed.

  Put it right: Treat the index as the changing counter.

• Using Σr² = [n(n + 1)/2]² instead of n(n + 1)(2n + 1)/6.

  Put it right: Memorise each standard result separately and check with n = 1.

Exam tips

• Write out the first two or three terms and the last term before summing, to confirm the pattern.
• When a sum starts at r = 0 or r = 2, adjust by adding or subtracting the missing terms explicitly.
• Check standard-result formulas with a small value of n, such as n = 1, before trusting them in a long calculation.