Recipe

Expand a sum or product written in index notation

Use when: you see a sigma or pi symbol with a lower limit, an upper limit, and a formula for the general term

Goal: the fully written-out list of terms, ready to add or multiply by hand

Taught in Summation and product notation

1. 1

   Note the index variable's name, its lower limit, its upper limit, and the formula written after the symbol.

   Example

   For \sum_{i=1}^{5}(2i-1): the index is i, the lower limit is 1, the upper limit is 5, and the general term is 2i-1.

   • the upper limit is at least the lower limit — there's at least one term; continue to Step 2.
   • the upper limit is below the lower limit — the sum or product is empty; skip straight to Step 3's empty-case rule.

   Why: every later step depends on knowing exactly which values to substitute and which expression to substitute them into.

2. 2

   Plug in every integer from the lower limit to the upper limit, one at a time, keeping each result as its own term.

   Example

   Substituting i=1,2,3,4,5 into 2i-1 gives 1, 3, 5, 7, 9.

   Why: this turns one compact symbol into the actual list of numbers that Step 3 combines.

3. 3

   Combine every term from Step 2 using the operation the symbol calls for.

   Example

   Adding the terms from Step 2: 1+3+5+7+9=25. For \prod_{i=1}^{4} i, multiply the terms 1,2,3,4 instead: 1\cdot2\cdot3\cdot4=24.

   • the symbol is \sum (sigma) — add every term together; an empty sum equals 0.
   • the symbol is \prod (pi) — multiply every term together; an empty product equals 1.

   ⚡ Shortcut

   When a sum runs over consecutive integers 1 through n with nothing else attached, skip unfolding entirely: \sum_{i=1}^{n} i = \frac{n(n+1)}{2}, the same formula the widget above checks numerically.

   Why: the two notations differ only in which operation combines the unfolded terms, since sigma starts its running total at 0 and pi starts at 1.