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The statement that the sum of all positive odd numbers up to 2n − 1 is a perfect square—more specifically, the perfect square n 2 —can be demonstrated by a proof without words. [3] In one corner of a grid, a single block represents 1, the first square. That can be wrapped on two sides by a strip of three blocks (the next odd number) to ...
The formula above is obtained by combining the composite Simpson's 1/3 rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and Simpson's 1/3 rule in the remaining subintervals. The result is then obtained by taking the mean of the two formulas.
Choose any odd square number k from this sequence (=) and let this square be the n-th term of the sequence. Also, let b 2 {\displaystyle b^{2}} be the sum of the previous n − 1 {\displaystyle n-1} terms, and let c 2 {\displaystyle c^{2}} be the sum of all n terms.
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},}
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
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.
The sum of two odd functions is odd. The difference between two odd functions is odd. The difference between two even functions is even. The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given domain.