Summasjoner

Summasjoner, or summations, are a fundamental operation in mathematics that express the addition of a sequence of terms. They are commonly written with the Greek letter Sigma: Σ from i = m to n of a_i, denoting the finite sum a_m + a_{m+1} + ... + a_n. For infinite series, the index extends to infinity, written Σ_{i=m}^∞ a_i, and the question is whether the sum converges to a finite value.

Finite sums have well-known closed forms in many cases. For example, Σ_{i=1}^n i = n(n+1)/2, and Σ_{i=1}^n i^2 = n(n+1)(2n+1)/6. Infinite sums are studied through their convergence properties. The partial sums S_N = Σ_{k=1}^N a_k approach a limit as N → ∞ if the series converges; otherwise, the series diverges. Common examples include the geometric series Σ_{k=0}^∞ ar^k = a/(1−r) for |r| < 1, and the harmonic series Σ_{k=1}^∞ 1/k, which diverges.

Key properties of summations include linearity: Σ (u_i + v_i) = Σ u_i + Σ v_i and Σ c_i u_i = (c_i) Σ u_i when constants can be factored. Reindexing and telescoping sums, where many terms cancel, are useful techniques for simplifying expressions like Σ (b_k − b_{k+1}) = b_1 − b_{n+1}.

Applications of summations span calculus, solving recurrence relations, probability, numerical analysis, and the development of series representations such as Fourier and power series. Understanding convergence criteria and standard test methods is essential when working with infinite sums.