sumn1N

sumn1N is a compact textual form for the finite summation of a sequence from n = 1 up to n = N. In standard notation this is written as S_N = ∑_{n=1}^N a_n, where a_n denotes the nth term of a given sequence. The expression represents the partial sum of the first N terms and is a central object in the study of finite series.

In the simplest case where a_n = n, the sum sumn1N corresponds to the sum of the first N natural numbers. This specific partial sum has the well-known closed form S_N = N(N+1)/2. For example, with N = 5, the sum is 15. Many other sequences have corresponding closed forms for their partial sums, though not all do; some require explicit calculation of each term or recognition of a telescoping structure.

Properties of finite sums include linearity: ∑_{n=1}^N (c·a_n) = c·∑_{n=1}^N a_n and ∑_{n=1}^N (a_n + b_n) = ∑_{n=1}^N a_n + ∑_{n=1}^N b_n. Partial sums also connect to infinite series, where the limit of S_N as N grows large (if it exists) defines the value of the series ∑_{n=1}^∞ a_n.

In computation, sumn1N can be evaluated by iteration, accumulation, or, when possible, by a closed-form formula for the chosen a_n. It is a common construct in mathematics, computer science, and related fields.