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