Series and Sigma Notation

Pre-Calculus · Axiom Academy

LESSON Series and Sigma Notation Turning a list of numbers into a sum — and writing that sum compactly with the symbol . Start with a sequence — a plain list such as . The matching series asks one question: what do we get when we add the whole list together? Watch the running total grow as each term is folded in — the series turns five numbers into a single number. A series adds them into one value 2. Sigma Notation: One Symbol for the Whole Sum Writing 2+4+6+8+10 is fine for five terms, but imagine a hundred. Sigma notation packs the entire sum into one symbol, , plus a recipe. It carries four parts: the index , the lower bound , the upper bound , and the expression . Press play to watch each part flag in, then see the machine unroll: the index ticks from 1 to 5 , each step drops the term 2i and adds it to the running total. The counter that changes each term — here i . (Any letter works: i , k , n , …) The starting value of the index, written below as i=1 . The ending value of the index, written above — here 5 . The formula to the right of — here 2i . Evaluate it at each index value. To expand sigma notation, substitute each integer from the lower to the upper bound into the expression and add: The lower bound need not be 1 — starting an index at 0 or any integer is perfectly legal, whichever makes the pattern cleanest. 3. Partial Sums Can Approach a Limit

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