Sum Of Floor Series

Evaluate 0 x e x d x.

Sum of floor series.

0 x. This is an application of the following fun result i published with natalio guersenzvaig a few years ago. The best strategy is to break up the interval of integration or summation into pieces on which the floor function is constant. Summation 8 formulas finite summation 7 formulas infinite summation 1 formula summation 8 formulas floor.

Ratio of mth and nth term in an arithmetic progression ap program to print gp geometric progression find m th summation of first n natural numbers. The other piece in the puzzle is the expression for the sum of an arithmetic sequence which can be found here. And floor n 2 2 floor n 2 1. Sum d j 1 k lfloor n d rfloor sum d lfloor n k rfloor 1 lfloor n j rfloor lfloor n d rfloor k lfloor n k rfloor j lfloor n j rfloor this is corollary 4 in some inversion formulas for sums of quotients.

At points of continuity the series converges to the true. Summation 8 formulas finite summation 7 formulas infinite summation 1 formula. Int limits 0 infty lfloor x rfloor e x dx. Sum of product of x and y such that floor n x y.

At points of discontinuity a fourier series converges to a value that is the average of its limits on the left and the right unlike the floor ceiling and fractional part functions. For y fixed and x a multiple of y the fourier series given converges to y 2 rather than to x mod y 0. Assuming n is even then floor n 2 floor n 1 2. Program to print arithmetic progression series.

Definite integrals and sums involving the floor function are quite common in problems and applications.