Alternatingsum

Alternatingsum, more commonly called an alternating sum, is the sum of a sequence in which consecutive terms enter with opposite signs. For a sequence a1, a2, a3,... the nth partial sum is s_n = a1 - a2 + a3 - a4 + ... + (-1)^{n-1} a_n. The associated infinite series is ∑_{n=1}^∞ (-1)^{n-1} a_n. When a_n ≥ 0, this is called an alternating series.

Special case: If a_n = n, the partial sums are 1 - 2 + 3 - 4 + 5 - 6 + ... . The even partial sums equal -n/2 and the odd partial sums equal (n+1)/2, illustrating how alternating sums can oscillate and grow in magnitude for non-convergent sequences.

Convergence: If a_n is nonnegative, decreasing to 0, the series converges (Leibniz criterion). The remainder after n terms is bounded by a_{n+1}.

Examples: The alternating harmonic series ∑ (-1)^{n-1} (1/n) converges to ln 2. The series 1 - 1 + 1 - 1 + ... does not converge in the usual sense, but can be assigned a value 1/2 by certain summation methods.

Generalizations: For any sequence a_n, the alternating form ∑ (-1)^{n-1} a_n is linear in the sense that sums can be distributed across terms, and convergence depends on the size and behavior of a_n.

Applications: Alternating sums appear in numerical analysis, series acceleration, and analytic evaluations in combinatorics and number theory.