Category Archives: calculus

The MacLaurin series for sin(x)

Posted on February 13, 2017 by Brent

In my previous post I said “recall the MacLaurin series for :” Since someone asked in a comment, I thought it was worth mentioning where this comes from. It would typically be covered in a second-semester calculus class, but it’s … Continue reading →

Posted in calculus, infinity, iteration | Tagged , , , , , , , | 4 Comments

Irrationality of pi: the integral that wasn't

Posted on February 11, 2010 by Brent

And now for the punchline! Today we’ll show that, for large enough values of , completing the proof of the irrationality of . First, let’s show that is positive when . We know that is positive for . But I … Continue reading →

Posted in algebra, calculus, convergence, famous numbers, proof, trig | Tagged , , , , | 8 Comments

Irrationality of pi: derivatives of f

Posted on December 20, 2009 by Brent

In my previous post in this series, we defined the function and showed that . Today we’ll show the surprising fact that, for every positive integer , although and are not necessarily zero, they are always integers. (The notation means … Continue reading →

Posted in calculus, famous numbers, proof | Tagged , , , , | 10 Comments

Irrationality of pi: the unpossible function

Posted on December 12, 2009 by Brent

Recall from my last post what we are trying to accomplish: by assuming that is a rational number, we are going to define an unpossible function! So, without further ado: Suppose , where and are positive integers. Define the function … Continue reading →

Posted in calculus, famous numbers, proof | Tagged , , , , | 7 Comments

Irrationality of pi

Posted on December 7, 2009 by Brent

Everyone knows that —the ratio of any circle’s diameter to its circumference—is irrational, that is, cannot be written as a fraction . This also means that ‘s decimal expansion goes on forever and never repeats …but have you ever seen … Continue reading →

Posted in calculus, famous numbers, proof | Tagged , , , | 14 Comments

Carnival of Mathematics #23: Haiku Edition

Posted on December 28, 2007 by Brent

Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be … Continue reading →

Posted in algebra, books, calculus, challenges, counting, fractals, geometry, infinity, links, meta, number theory, pascal's triangle, pattern, people, sequences, trig, video | 17 Comments

The Most Beautiful Equation in the World

Posted on April 13, 2007 by Brent

This post is a special shout-out to my former students who are now taking calculus (if you don’t know any calculus, just hang tight… there will be more calculus-less math goodness coming your way soon). This post is 100% money-back … Continue reading →

Posted in calculus, famous numbers, proof | 46 Comments