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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.
Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence f n we define the two sequences of sums := = ~:= = (+) (+) (+), for all n ≥ 0, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.
However, certain calculations may result in the program attempting to evaluate an infinite number of elements; for example, requesting the length of the list or trying to sum the elements of the list with a fold operation would result in the program either failing to terminate or running out of memory.
If the sum is of the form = ()where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...