Mathematical Methods for Physics and Engineering - Matematica.NET

4.3 CONVERGENCE OF INFINITE SERIESUsing the integral test, we consider∫ N( )1N1−plim dx = lim ,N→∞ xp N→∞ 1 − pand it is obvious that the limit tends to zero for p>1andto∞ for p ≤ 1.Cauchy’s root testCauchy’s root test may be useful in testing for convergence, especially if the nthterms of the series contains an nth power. If we define the limitρ = lim (u n ) 1/n ,n→∞then it may be proved that the series ∑ u n converges if ρ1 then theseries diverges. Its behaviour is undetermined if ρ =1.◮Determine whether the following series converges:∞∑( ) n 1=1+ 1 n 4 + 1 27 + ··· .n=1Using Cauchy’s root test, we findand hence the series converges. ◭( ) 1ρ = lim =0,n→∞ nGrouping termsWe now consider the Riemann zeta series, mentioned above, with an alternativeproof of its convergence that uses the method of grouping terms. In general thereare better ways of determining convergence, but the grouping method may beused if it is not immediately obvious how to approach a problem by a bettermethod.First consider the case where p>1, and group the terms in the series asfollows:S N = 1 ( 11 p + 2 p + 1 ) ( 13 p +4 p + ···+ 1 )7 p + ··· .Now we can see that each bracket of this series is less than each term of thegeometric seriesS N = 1 1 p + 2 2 p + 4 4 p + ··· .This geometric series has common ratio r = ( 12) p−1;sincep>1, it follows thatr1.129