Tag Archives: talk

Notes from a talk on the Mean Value Theorem

1. Introduction

When I first learned the Mean Value Theorem and the Intermediate Value Theorem, I thought they were both intuitively obvious and utterly useless. In one of my courses in analysis, I was struck when, after proving the Mean Value Theorem, my instructor said that all of calculus was downhill from there. But it was a case of not being able to see the forest for the trees, and I missed the big picture.

I have since come to realize that almost every major (and often, minor) result of calculus is a direct and immediate consequence of the Mean Value Theorem and the Intermediate Value Theorem. In this talk, we will focus on the forest, the big picture, and see the Mean Value Theorem for what it really is: the true Fundamental Theorem of Calculus.


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Friendly Introduction to Sieves with a Look Towards Progress on the Twin Primes Conjecture

This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.’ During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I’ll provide missing details and/or sources for additional reading here.

Furthermore, I like this talk, so I think it’s worth preserving.

1. Introduction

We talk about sieves and primes. Long, long ago, Euclid famously proved the infinitude of primes ($latex {\approx 300}$ B.C.). Although he didn’t show it, the stronger statement that the sum of the reciprocals of the primes diverges is true:

$latex \displaystyle \sum_{p} \frac{1}{p} \rightarrow \infty, $

where the sum is over primes.

Proof: Suppose that the sum converged. Then there is some $latex {k}$ such that

$latex \displaystyle \sum_{i = k+1}^\infty \frac{1}{p_i} < \frac{1}{2}. $

Suppose that $latex {Q := \prod_{i = 1}^k p_i}$ is the product of the primes up to $latex {p_k}$. Then the integers $latex {1 + Qn}$ are relatively prime to the primes in $latex {Q}$, and so are only made up of the primes $latex {p_{k+1}, \ldots}$. This means that

$latex \displaystyle \sum_{n = 1}^\infty \frac{1}{1+Qn} \leq \sum_{t \geq 0} \left( \sum_{i > k} \frac{1}{p_i} \right) ^t < 2, $

where the first inequality is true since all the terms on the left appear in the middle (think prime factorizations and the distributive law), and the second inequality is true because it’s bounded by the geometric series with ratio $latex {1/2}$. But by either the ratio test or by limit comparison, the sum on the left diverges (aha! Something for my math 100 students), and so we arrive at a contradiction.

Thus the sum of the reciprocals of the primes diverges. $latex \diamondsuit$


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