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# Category Archives: calculus

## The MacLaurin series for sin(x)

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

## Irrationality of pi: the integral that wasn't

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 inequality, integral, irrational, Niven, pi
8 Comments

## Irrationality of pi: derivatives of f

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 derivatives, irrationality, Ivan Niven, pi, proof
10 Comments

## Irrationality of pi: the unpossible function

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 irrational, Ivan Niven, pi, proof, symmetric
7 Comments

## Irrationality of pi

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

## The Most Beautiful Equation in the World

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