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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: [1] ¯ = ().
The sum of a finite initial segment of an infinite geometric series is called a finite geometric series, that is: [2] + + + + + = =. When r > 1 {\displaystyle r>1} it is often called a growth rate or rate of expansion.
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: (). The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter h increases is sometimes called a spectral-element method. Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems.
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus .
Instead, this tangent is estimated by using the original Euler's method to estimate the value of () at the midpoint, then computing the slope of the tangent with (). Finally, the improved tangent is used to calculate the value of + from . This last step is represented by the red chord in the diagram.
The next step is to multiply the above value by the step size , which we take equal to one here: = = Since the step size is the change in , when we multiply the step size and the slope of the tangent, we get a change in value.