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This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. It can be used in conjunction with other tools for evaluating sums.
The sum over all residues ensures that adjustments are uniformly applied across all possible offsets within each block of terms. This uniform distribution of the "correction" across different intervals defined by x − r {\displaystyle x-r} functions similarly to telescoping over a very large sequence.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions , vectors , matrices , polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
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1. Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation = means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of F.