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Re: Re: Help on Infinite Series
*To*: mathgroup at smc.vnet.net
*Subject*: [mg2477] Re: [mg2451] Re: Help on Infinite Series
*From*: Robert Pratt <rpratt at math.unc.edu>
*Date*: Sun, 12 Nov 1995 22:29:22 -0500
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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