sumk0N1

Sumk0N1 is a compact informal name for the finite sum with lower bound k = 0 and upper bound k = N − 1. In mathematics this is written as sum_{k=0}^{N-1} f(k), where f is a function defined on the nonnegative integers up to N−1. The term "sumk0N1" appears in notes or programming contexts as a shorthand for this summation range without writing the full expression.

Definition and scope: Let N be a nonnegative integer and f be defined on {0,1,...,N−1}. The sum equals the aggregate of f(k) as k runs from 0 to N−1. If N = 0, the index set is empty and the sum is defined as 0. This finite sum is a fundamental construct in discrete mathematics, combinatorics, and computer science.

Key properties: The summation operator is linear: sum_{k=0}^{N-1} [a f(k) + b g(k)] = a sum_{k=0}^{N-1} f(k) + b sum_{k=0}^{N-1} g(k). Index shifting allows reindexing: sum_{k=0}^{N-1} f(k) = sum_{j=0}^{N-1} f(j). Also, closed forms exist for many simple f. For example, sum_{k=0}^{N-1} 1 = N, sum_{k=0}^{N-1} k = N(N−1)/2, and sum_{k=0}^{N-1} k^2 = (N−1)N(2N−1)/6.

Geometric examples: sum_{k=0}^{N-1} r^k = (1 − r^N)/(1 − r) for r ≠ 1. In algorithms and analysis, these sums help estimate running times, derive series identities, and simplify recurrences. When N is large, a closed form is usually preferable to iterative summation.

See also: finite sum, summation notation, arithmetic series, geometric series, index notation, discrete mathematics.