Sumregler

Sumregler, literally “sum rules,” is a term used in mathematics to describe the rules that govern the operation of summation. The concept covers finite sums (manual addition), linear combinations of sequences, and the use of summation notation (sigma). Core rules include linearity of the sum: for sequences a_i and b_i and scalar c, sum_i (a_i + b_i) = sum_i a_i + sum_i b_i, and sum_i (c a_i) = c sum_i a_i. These lead to the ability to distribute summation over addition and factor constants out of sums.

In calculus, sum rules extend to derivatives and integrals: the derivative of a sum equals the sum of the derivatives, and the integral of a sum equals the sum of the integrals (up to a constant for indefinite integrals). In probability, the linearity of expectation is a fundamental sum rule: E[X+Y] = E[X] + E[Y] and for constants, E[cX] = c E[X].

For infinite series, sum rules distinguish finite sums from infinite sums; convergence determines when limits can be interchanged or when a series has a well-defined value. Basic geometric and harmonic series illustrate common forms. More advanced sum rules address exchanging sums with limits (Fubini/Tonelli-type theorems) and term-by-term manipulation of power series.

Sumregler are used across algebra, analysis, and applied fields to simplify calculations and structure proofs. They underpin computational techniques and theoretical results alike. See also: summation, sigma notation, series, linearity of sums.