Mathematical Methods for Physics and Engineering - Matematica.NET

4.3 CONVERGENCE OF INFINITE SERIESRatio comparison testAs its name suggests, the ratio comparison test is a combination of the ratio andcomparison tests. Let us consider the two series ∑ u n and ∑ v n and assume thatwe know the latter to be convergent. It may be shown that ifu n+1u n≤ v n+1v nfor all n greater than some fixed value N then ∑ u n is also convergent.Similarly, ifu n+1≥ v n+1u n v nfor all sufficiently large n, and ∑ v n diverges then ∑ u n also diverges.◮Determine whether the following series converges:∞∑ 1(n!) =1+ 1 2 2 + 1 2 6 + ··· .2n=1In this case the ratio of successive terms, as n tends to infinity, is given by[ ] 2 ( ) 2n!1R = lim= lim ,n→∞ (n +1)! n→∞ n +1which is less than the ratio seen in (4.11). Hence, by the ratio comparison test, the seriesconverges. (It is clear that this series could also be found to be convergent using the ratiotest.) ◭Quotient testThe quotient test may also be considered as a combination of the ratio andcomparison tests. Let us again consider the two series ∑ u n and ∑ v n , and defineρ as the limit( )unρ = lim . (4.12)n→∞ v nThen, it can be shown that:(i) if ρ ≠ 0 but is finite then ∑ u n and ∑ v n either both converge or bothdiverge;(ii) if ρ = 0 and ∑ v n converges then ∑ u n converges;(iii) if ρ = ∞ and ∑ v n diverges then ∑ u n diverges.127