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MathGroup Archive 1990

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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


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