Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSwhich is merely the series obtained by setting x = 1 in the Maclaurin expansion of exp x(see subsection 4.6.3), i.e.exp(1) = e =1+ 1 1! + 1 2! + 1 3! + ··· .Clearly this second series is convergent, since it consists of only positive terms and has afinite sum. Thus, since each term u n in the series (4.7) is less than the corresponding term1/n! in (4.8), we conclude from the comparison test that (4.7) is also convergent. ◭D’Alembert’s ratio testThe ratio test determines whether a series converges by comparing the relativemagnitude of successive terms. If we consider a series ∑ u n and set( )un+1ρ = lim , (4.9)n→∞ u nthen if ρ1 the series is divergent; if ρ =1then the behaviour of the series is undetermined by this test.To prove this we observe that if the limit (4.9) is less than unity, i.e. ρ