In this post I’m going to discuss two “proofs” by induction. That is, they’ll both be attempts at proving a statement, but they’ll both have flaws in their arguments. My hope is to alert people to potential errors when attempting to prove a statement by induction. Claim 1: For all n, . “Proof”: Let denote that claim that . Suppose that is true for some k; that is, suppose . Now, is the claim that . If is true, then , which proves that the truth of implies the truth of . By the principle of mathematical induction, then, is true for all n; that is, for all n. Claim 2: For all , all people ..read more

A few months ago Mathematics Magazine published a paper of mine, “A Combinatorial View of Sums of Powers.” In it I give a combinatorial interpretation for the power sum , together with combinatorial proofs of two formulas for this power sum. (An earlier version of some of the results in this paper actually appeared in a blog post from several years ago.) Recently, however, I received an email from Michael Maltenfort at Northwestern University, pointing out a couple of errors in the paper. I’m recording those errors here. The most serious mistake Maltenfort pointed out to me is the paper’s clai ..read more

The Riemann zeta function can be expressed as , for complex numbers s whose real part is greater than 1. By analytic continuation, can be extended to all complex numbers except where . The power sum is given by . (Sometimes M is given as the upper bound, but for this post it’s more convenient to use .) Recently I learned of a really nice relationship between the zeta function and the power sum for . It’s due to Mináč [1], and I reproduce his proof here. Theorem: For positive integers a, . Now, of course M in is a positive integer, so what do we mean by integrating from 0 to 1? Well, it ..read more

One of the most important theoretical results in linear programming is that every LP has a corresponding dual program. Where, exactly, this dual comes from can often seem mysterious. Several years ago I answered a question on a couple of Stack Exchange sites giving an intuitive explanation for where the dual comes from. Those posts seem to have been appreciated, so I thought I would reproduce my answer here. Suppose we have a primal problem as follows. Now, suppose we want to use the primal’s constraints as a way to find an upper bound on the optimal value of the primal. If we multiply the fi ..read more