Re: Re: Newbie Limit problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg53484] Re: [mg53471] Re: [mg53439] Newbie Limit problem*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 14 Jan 2005 08:54:29 -0500 (EST)*References*: <200501120841.DAA09647@smc.vnet.net> <200501130812.DAA03805@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Unfortunately (as often happens) in my posting there was a certian disharmony between the words and the deeds ;-) It was not all "input form". In particular > d+2 =B9 should have been d + 2*Pi and > Abs[d-2 =B9] should have been: Abs[b-2*Pi] Andrzej On 13 Jan 2005, at 09:12, Andrzej Kozlowski wrote: > On 12 Jan 2005, at 09:41, Ken Tozier 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]]\) >> >> 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. >> >> \!\(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? >> >> Thanks >> >> Ken >> > > You are indeed doing a few things wrong but they are not responsible > for the lack of result. The first very bad thing you are doing is > writing 0.5 for the power exponent 1/2. In Mathematica these two things > > (0.5 and 1/2) are quite different and using the former in non-numerical > > problems can cause all sorts of weird problems. Secondly, your > assumptions are obviously not very helpful. For a start, what is the > use of telling Mathematica that s is an integer if you also do not > include the assumption that k is also an integer? But in any case even > > if you add that information it will make any difference; in fact > telling Mathematica that a parameter is an integer in most cases only > reduces the number of transformations it can perform and makes it less > > rather than more likely it can solve the problem. > > But actually, the main points seem to be: > > 1) Mathematica just can't do this for general d > > 2) While it is easy to prove that the sum is convergent (see below) do > > you have any special reason to expect that there is an explicit > "closed" formula for it? Such closed formulas are actually quite rare > so unless you are lucky neither Mathematica nor anyone else will find > one. > > However, there is quite a lot in this connection with this problem that > > Mathematica can be helpful with. In fact we can get quite reasonable > bounds for the value of the limit. For convenience I shall assume that > > d>0. > First of all, let's define a function of two variables: > > > p[s_,d_]:=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^2], {k, 0, s}] > > Note that I am using InputForm, which makes it easy to paste > expressions into Mathematica in a readable way. > > To start, observe that while Mathematica can't find the general limit > Limit[p[s,d],s->Infinity], it can find it for the special value d=0: > > > Limit[p[s, 0], s -> Infinity] > > 2*Pi > > In general Mathematica can't find any closed form expression for the > limit Limit[p[s, d], s -> Infinity]. It is easy to see that the > difficulty is caused by the term d*(Cos[(2*k*Pi)/s] - Cos[(2*(1 + > k)*Pi)/s]). Fortunately it is rather easy to find bounds for this term: > > First, d*(Cos[(2*k*Pi)/s] - Cos[(2*(1 + k)*Pi)/s]) can be re-written as > > > Simplify /@ TrigFactor[d*(Cos[(2*k*Pi)/s] - Cos[(2*(1 + k)*Pi)/s])] > > > 2*d*Sin[Pi/s]*Sin[(Pi + 2*k*Pi)/s] > > We see that this always between -2d Pi/s and 2d Pi/s . This means that > > our limit will lie between > > > Simplify[Limit[Sum[Sqrt[(d^2+2*s^2-2* > s*(s*Cos[(2*Pi)/s]-2*Pi (d/s)))/s^2],{k,0,s}],s->Infinity],d>0] > > d+2 =B9 > > and > > > Simplify[Limit[Sum[Sqrt[(d^2+2*s^2-2* > s*(s*Cos[(2*Pi)/s]+2*Pi (d/s)))/s^2],{k,0,s}],s->Infinity],d>0] > > > Abs[d-2 =B9] > > > Let's try checking this for some random values of d, e.g. d=5 and d= > > 23. Let's use a large value of s, say > > s=10000; > putting > d=5; > > we obtain > N[p[s,d]] > > 7.32662 > > The upper bound gives: > > N[d+2 =B9] > > 11.2832 > > while the lower bouund gives: > > N[Abs[d-2 =B9]] > > > 1.28319 > > The average of the two is 6.2832, not that far off. > > let's try > > d=23; > > > N[p[s,d]] > > > 23.4335 > > > N[d+2 =B9] > > > 29.2832 > > > N[Abs[d-2 =B9]] > > > 16.7168 > > This time the average is 23, which is quite good. > > > > Andrzej Kozlowski > Chiba, Japan > http://www.akikoz.net/~andrzej/ > http://www.mimuw.edu.pl/~akoz/ >

**References**:**Newbie Limit problem***From:*Ken Tozier <kentozier@comcast.net>

**Re: Newbie Limit problem***From:*Andrzej Kozlowski <akozlowski@gmail.com>