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