Limit[Sum[...,{n,1,Infinity}], n->Infinity]

*To*: mathgroup at yoda.ncsa.uiuc.edu*Subject*: Limit[Sum[...,{n,1,Infinity}], n->Infinity]*From*: "Mr. Mathematica Man" <LWRIGHT at HMCVAX.CLAREMONT.EDU>*Date*: Fri, 2 Nov 1990 18:10 PST

Here is my dilemma. I am trying to model things like Reimann sums and other techniques for approximating the area underneath a curve. Here is a sample scenario. | | |----|----|----|----|----|----|----|----| 0 1 2 We have some interval (in this case, [0,2]) that we are concerned with. It is divided into n sections. Say we are interested in the function f(x) = x^3. So we want to find the area under the graph of x^3 from x=0 to 2. If we subdivide the interval [0,2] into n equal subintervals, then dx = 2/n and xi = 0 + i(2/n) = 2i/n Therefore, Sum[f(xi)dx,{i,1,n}] = Sum[(xi)^3 dx, {i,1,n}] = Sum[(2i/n)^3 (2/n), {i,1,n}] = 16/(n^4) Sum[i^3,{i,1,n}] *** The problem is that Mathematica cannot proceed from this last step. It also doesn't seem like it can factor out irrelevant terms. Mathematica gives me the following: Sum[(16 i^3)/(n^4),{i,1,n}] *** The BIGGER problem is that we are unable to get a limit of this sum as n goes to infinity. If you attempt to do it, Mathematica will give you several screens full of garbage. Induction tells us that it is ((n^2)(n+1)^2)/4. Consequently our limit is 4. Is there some way to handle limits of infinite sums? Is there some way to use a lookup table for special cases? What is Mathematica actually doing when it is computing the limit and/or sums? What about numerically approximating? Any help would be greatly appreciated since so many things are represented as a limit of an infinite sum. Thanks, Lyle Wright Mathematica Developer Harvey Mudd College Claremont, CA 91711 lwright at hmcvax.claremont.edu