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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.
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.
The exponential function is the sum of the power series [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every x {\displaystyle x} per the ratio test .
Process B transforms the sum involving f into one involving a function g defined in terms of the derivative of f. Suppose that f' is monotone increasing with f'(a) = α, f'(b) = β. Then f' is invertible on [α,β] with inverse u say. Further suppose f'' ≥ λ > 0. Write
The exponential function e x (in blue), and the sum of the first n + 1 terms of its Taylor series at 0 (in red). The exponential function e x {\displaystyle e^{x}} (with base e ) has Maclaurin series [ 12 ]
The exponential function (in blue), and its improving approximation by the sum of the first n + 1 terms of its Maclaurin power series (in red). So So n=0 gives f ( x ) = 1 {\displaystyle f(x)=1} ,
Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.
This last non-simple continued fraction (sequence A110185 in the OEIS), equivalent to = [;,,,,,...], has a quicker convergence rate compared to Euler's continued fraction formula [clarification needed] and is a special case of a general formula for the exponential function: