Sum Of Floor Series

Assuming n is even then floor n 2 floor n 1 2.

Sum of floor series.

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. And floor n 2 2 floor n 2 1. At points of continuity the series converges to the true. Sum of product of x and y such that floor n x y.

Definite integrals and sums involving the floor function are quite common in problems and applications. Sum of elements of a geometric progression gp in a given range. 0 x. 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.

This is an application of the following fun result i published with natalio guersenzvaig a few years ago. Laguerre based hermite bernoulli polynomials associated with bilateral series ahmad khan waseem araci serkan acikgoz mehmet and esi ayhan tbilisi mathematical journal 2018 airflow and heat transfer in the slot vented room with radiant floor heating unit liu xiang long gong guang cai cheng heng sheng and ding li xing journal of. The other piece in the puzzle is the expression for the sum of an arithmetic sequence which can be found here. The best strategy is to break up the interval of integration or summation into pieces on which the floor function is constant.

Evaluate 0 x e x d x. Summation 8 formulas finite summation 7 formulas infinite summation 1 formula. Sum of sum series of first n natural numbers sum of series formed by difference between product and sum of n natural numbers program to find sum of series 1 1 2 1 3 1 4. 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.

Int limits 0 infty lfloor x rfloor e x dx.