Nov 28, 2014 · Want to improve this question? Guide the asker to update the question so it focuses on a single, specific problem. Narrowing the question will help others answer the …

Formulation Main article: Writing a Proof by Induction Now that we've gotten a little bit familiar with the idea of proof by induction, let's rewrite everything we learned a little more formally. Proof …

A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a …

The Lagrange interpolation formula is a way to find a polynomial which takes on certain values at arbitrary points. Specifically, it gives a constructive proof of the theorem below. This theorem …

Proof assuming M M has entries in C C Suppose M M is an n n -by- n n matrix. When M M has entries in C C, one can prove the Cayley-Hamilton theorem as follows: A matrix M ∈ M n (C) M …

Proof of Strong Induction This proof is almost identical to the proof of standard induction. Can you spot the differences? Let S S be a set of positive integers with the following properties: The …

Proof of Power Rule 1: Using the identity x c = e c ln x, xc = eclnx, we differentiate both sides using derivatives of exponential functions and the chain rule to obtain

The Riemann zeta function for s ∈ C s ∈ C with Re (s)> 1 Re(s)> 1 is defined as ζ (s) = ∑ n = 1 ∞ 1 n s ζ (s) = n=1∑∞ ns1. It is then defined by analytical continuation to a meromorphic function …

Bijective Functions A common proof technique in combinatorics, number theory, and other fields is the use of bijections to show that two expressions are equal. To prove a formula of the form …

Proof Suppose we have a sequence of length n n, and we want the number of ways to choose k k elements. Obviously, there are n n choices for the first element. There are n 1 n−1 ways to …