From: harper@kauri.vuw.ac.nz (John Harper) Newsgroups: sci.math Subject: Re: e in the calculus class Date: 9 Oct 1998 02:18:40 GMT In article <6vjkcs$b66$1@nnrp1.dejanews.com>, wrote: >In article <361D16C0.41C6@lie.math.missouri.edu>, > Michael Wodzak wrote: > >Consider f(x) = (1+x/n)^n > >Then f'(x) = (1+x/n)^(n-1) > >So f'(x)/f(x) = 1+x/n which becomes 1 in the limit as n->infinity. Define >exp(x) to be the limit of f(x) as n->infinity and define e = exp(1). You >should now be able to fill in the details... Simple-minded fluid mechanicals like me know that that sort of thing needs extreme care. I like many others believed Proudman & Johnson (J Fluid Mech 12 161-168 1962) until van Dommelen & Shen (J Fluid Mech 157 1-16 1985) revealed an error. The point at issue was that if D(t) is a function obeying (dD/dt)/D -> 0 as t -> infinity, then P&J had assumed D -> constant, but vD&S pointed out that D ~ any power of t would also satisfy the asymptotic condition, (and that is not the general solution either). Other features of the original problem (too long to give here; see vD&S) led to D ~ 1/sqrt(t). Warning to PhD supervisors: read your students' stuff with care. D W Moore's thesis (1958) was supervised by Proudman, and it could have prevented the error being made! John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)471 5341 fax (+64)(4)495 5045