Closed-Form Sum of an Arithmetic Series in Algebra
An arithmetic series is a sum whose terms increase by a constant step, and such a sum has a closed form (for the first k positive integers, k(k+1)/2) that replaces an unevaluated summation with a single expression. It also covers shifting the starting index by subtracting the sum of the excluded lower terms. After learning this a student can turn a sum written with a sigma into an explicit polynomial in its upper limit, and adjust that formula when the sum does not begin at the series' natural first term.
Closed-Form Formulas for Summations in Calculus
In the study of summations, an open form expresses a sum via an explicit or implicit ellipsis (listing terms up to an unspecified final term), whereas a closed form expresses the same sum as a single…