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MathGroup Archive 1995

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Re: Re: Help on Infinite Series

  • Subject: [mg2477] Re: [mg2451] Re: Help on Infinite Series
  • From: rpratt at math.unc.edu (Robert Pratt)
  • Date: Mon, 13 Nov 1995 03:29:22 GMT
  • Approved: usenet@wri.com
  • Distribution: local
  • Newsgroups: wri.mathgroup
  • Organization: Wolfram Research, Inc.
  • Sender: daemon at wri.com ( )

While Bob Hanlon's solution seems convincing, especially since it "agrees 
with published results," a rigorous proof by mathematical induction is 
necessary in order to remove any doubts.  This method of proof is useful 
when one wants to demonstrate that a statement is true for all integers.  I 
don't know if Matt Fisher's question was about programming or about
mathematics, but if he wants to prove that the results hold, "proof by 
Mathematica" will not suffice.  Here are the proofs for the sum of the 
first n squares and the sum of the first n cubes.

Prove: Sum[k^2, {k, 1, n}] = n (n+1) (2n+1) /6

First verify that the result holds for the first case n=1:

Sum[k^2, {k, 1, 1}] = 1^2 = 1 = 1 (1+1) (2(1)+1) /6

Now assume the result holds for the first m squares and use this 
assumption to show the result holds for the first m+1 squares:

Assume: Sum[k^2, {k, 1, m}] = m (m+1) (2m+1) /6

Then Sum[k^2, {k, 1, m+1}] = m (m+1) (2m+1) /6 + (m+1)^2

	= (m (2m+1) (m+1) + 6(m+1)^2) /6

	= (m+1) (m (2m+1)+6(m+1)) /6

	= (m+1) (2m^2+7m+6) /6

	= (m+1) (2m+3) (m+2) /6

	= (m+1) ((m+1)+1) (2(m+1)+1) /6

QED

Prove: Sum[k^3, {k, 1, n}] = n^2 (n+1)^2 /4

First verify that the result holds for the first case n=1:

Sum[k^3, {k, 1, n}] = 1^3 = 1 = 1^2 (1+1)^2 /4

Now assume the result holds for the first m cubes and use this assumption 
to show the result holds for the first m+1 cubes:

Assume: Sum[k^3, {k, 1, m}] = m^2 (m+1)^2 /4

Then Sum[k^3, {k, 1, m+1}] = m^2 (m+1)^2 /4 + (m+1)^3

	= (m^2 (m+1)^2 + 4(m+1)^3) /4

	= (m+1)^2 (m^2 + 4(m+1)) /4

	= (m+1)^2 (m+2)^2 /4

	= (m+1)^2 ((m+1)+1)^2 /4

QED

Note that these proofs do not give what the sum should be.  Instead, you 
must "know" what the sum is before you start.  This idea is similar to 
that encountered in first-semester calculus when one wants to use the 
delta-epsilon definition to prove that a limit has a certain value.  You 
must "know" what the limit is before you start.  In this sense, Bob 
Hanlon's program is very useful for "finding" the sums, i.e. generating 
conjectures.  Perhaps this search process was more what Matt was looking 
for.  Nevertheless, the conjectures must be proved as above.

Rob Pratt
Department of Mathematics
The University of North Carolina at Chapel Hill
CB# 3250, 331 Phillips Hall
Chapel Hill, NC  27599-3250

rpratt at math.unc.edu





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