Mathematical Methods for Physics and Engineering - Matematica.NET

4.2 SUMMATION OF SERIES4.2.1 Arithmetic seriesAn arithmetic series has the characteristic that the difference between successiveterms is constant. The sum of a general arithmetic series is writtenN−1∑S N = a +(a + d)+(a +2d)+···+ [a +(N − 1)d] = (a + nd).Rewriting the series in the opposite order and adding this term by term to theoriginal expression for S N , we findS N = N 2 [a + a +(N − 1)d] = N (first term + last term). (4.2)2If an infinite number of such terms are added the series will increase (or decrease)indefinitely; that is to say, it diverges.n=0◮Sum the integers between 1 and 1000 inclusive.This is an arithmetic series with a =1,d =1andN = 1000. Therefore, using (4.2) we findS N = 1000 (1 + 1000) = 500500,2which can be checked directly only with considerable effort. ◭4.2.2 Geometric seriesEquation (4.1) is a particular example of a geometric series, which has thecharacteristic that the ratio of successive terms is a constant (one-half in thiscase). The sum of a geometric series is in general writtenN−1∑S N = a + ar + ar 2 + ···+ ar N−1 = ar n ,where a is a constant and r is the ratio of successive terms, the common ratio. Thesum may be evaluated by considering S N and rS N :n=0S N = a + ar + ar 2 + ar 3 + ···+ ar N−1 ,rS N = ar + ar 2 + ar 3 + ar 4 + ···+ ar N .If we now subtract the second equation from the first we obtainand hence(1 − r)S N = a − ar N ,S N = a(1 − rN ). (4.3)1 − r117