Mathematical Methods for Physics and Engineering - Matematica.NET

4.2 SUMMATION OF SERIES◮Sum the seriesS =2+ 5 2 + 8 2 2 + 112 3 + ··· .This is an infinite arithmetico-geometric series with a =2,d =3andr =1/2. Therefore,from (4.5), we obtain S = 10. ◭4.2.4 The difference methodThe difference method is sometimes useful in summing series that are morecomplicated than the examples discussed above. Let us consider the general seriesN∑u n = u 1 + u 2 + ···+ u N .n=1If the terms of the series, u n , can be expressed in the formu n = f(n) − f(n − 1)for some function f(n) then its (partial) sum is given byN∑S N = u n = f(N) − f(0).n=1This can be shown as follows. The sum is given byS N = u 1 + u 2 + ···+ u Nand since u n = f(n) − f(n − 1), it may be rewrittenS N =[f(1) − f(0)] + [f(2) − f(1)] + ···+[f(N) − f(N − 1)].By cancelling terms we see thatS N = f(N) − f(0).◮Evaluate the sumN∑n=11n(n +1) .Using partial fractions we find( 1u n = −n +1 − 1 ).nHence u n = f(n) − f(n − 1) with f(n) =−1/(n +1),andsothesumisgivenbyS N = f(N) − f(0) = − 1N +1 +1= NN +1 . ◭119