Sumk1i

Sumk1i denotes a finite sum in which the index k runs from 1 to i of a sequence a_k. In standard notation this is written as sum_{k=1}^i a_k. The value depends on both the upper limit i and the terms a_k; it is often called the i-th partial sum of {a_k}.

If i is fixed, the sum is a single number; if i is treated as a variable, the expression defines a function S(i) = sum_{k=1}^i a_k, known as the partial sum function.

Examples help illustrate the concept. For a_k = k, sum_{k=1}^i k = i(i+1)/2. For a_k = 1, sum_{k=1}^i 1 = i. For a_k = 2^k, sum_{k=1}^i 2^k = 2^{i+1}-2. In general, the sum of an arbitrary sequence a_k does not always have a closed form; it depends on the specific terms involved.

Key properties of sumk1i include linearity: sum_{k=1}^i (a_k + b_k) = sum_{k=1}^i a_k + sum_{k=1}^i b_k and sum_{k=1}^i c a_k = c sum_{k=1}^i a_k for any constant c. If a_k = b_k - b_{k-1}, the sum telescopes to b_i - b_0, illustrating how some sums collapse.

Applications of partial sums are widespread in mathematics and computer science. They form the basis of series analysis, discrete mathematics, and algorithms that use prefix sums to aggregate data. When i increases without bound, the infinite series sum_{k=1}^\infty a_k converges if the limit of S(i) exists. See also sigma notation, arithmetic series, geometric series, and prefix sums.