Re: Newbie Limit problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg53473] Re: Newbie Limit problem*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Thu, 13 Jan 2005 03:12:22 -0500 (EST)*Organization*: The University of Western Australia*References*: <cs2olb$9ni$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cs2olb$9ni$1 at smc.vnet.net>, Ken Tozier <kentozier at comcast.net> wrote: > I'm trying to get a limit for a sum that I know converges as s->infinity > > \!\(Limit[\[Sum]\+\(k = 0\)\%s\((\(d\^2 + 2\ s\^2 - 2\ s\ \((s\ > Cos[\(2\ \ > \[Pi]\)\/s] + d\ \((Cos[\(2\ k\ \[Pi]\)\/s] - Cos[\(2\ \((1 + k)\)\ \ > \[Pi]\)\/s])\))\)\)\/s\^2)\)\^0.5`, s \[Rule] \[Infinity]]\) > > but all I'm getting for a result is the exact expression I plug into > the Limit function. > > \!\(Limit[\[Sum]\+\(k = 0\)\%s\((\(d\^2 + 2\ s\^2 - 2\ s\ \((s\ > Cos[\(2\ \ > \[Pi]\)\/s] + d\ \((Cos[\(2\ k\ \[Pi]\)\/s] - Cos[\(2\ \((1 + k)\)\ \ > \[Pi]\)\/s])\))\)\)\/s\^2)\)\^0.5`, s \[Rule] \[Infinity]]\) And you are suprised by this? Some observations: [1] x^0.5 is not the same as x^(1/2) (or Sqrt[x]) [2] The factor 1/s^2 in the denominator of the Sqrt can be factored out. [3] What sort of functional form do you expect as s->infinity? If you compute the sum for fixed s, say s=3, you will see that form of the expression is quite complicated FullSimplify[With[{s = 3}, Sum[Sqrt[ (d^2 + 2 s^2 - 2 s (s Cos[(2 Pi)/s] + d (Cos[(2 k Pi)/s] - Cos[(2 (1 + k) Pi)/s])))]/s, {k, 0, s}]], d > 0] Try this for s = 7. Then try and guess the form as s -> Infinity. > I read the documentation which describes this scenario like so: "Limit > returns unevaluated when it encounters functions about which it has no > specific information. Limit therefore by default makes no explicit > assumptions about symbolic functions." > > Next I tried to explicitly give it some assumptions like so: > > \!\(Assuming[{s \[Element] Integers, d \[Element] Reals}, > Limit[\[Sum]\+\(k = 0\)\%s\((\(1\/s\^2\) \((d\^2 + 2\ s\^2 - 2\ s\ > \((s\ \ > Cos[\(2\ \[Pi]\)\/s] + d\ \((Cos[\(2\ k\ \[Pi]\)\/s] - Cos[\(2\ \((1 + > k)\)\ \ > \[Pi]\)\/s])\))\))\))\)\^0.5`, s \[Rule] \[Infinity]]]\) > > which yields the same result. Which is not suprising. These assumptions do not help in reducing an intractable sum. > > \!\(Limit[\[Sum]\+\(k = 0\)\%s\((\(d\^2 + 2\ s\^2 - 2\ s\ \((s\ > Cos[\(2\ \ > \[Pi]\)\/s] + d\ \((Cos[\(2\ k\ \[Pi]\)\/s] - Cos[\(2\ \((1 + k)\)\ \ > \[Pi]\)\/s])\))\)\)\/s\^2)\)\^0.5`, s \[Rule] \[Infinity]]\) > > Am I using "Limit" wrong? Or is there some other way to write the > expression to get Mathematica to give me the limit? Why do you think Mathematica should be able to compute the limit? Cheers, Paul -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul