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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 secondsemester 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 shoutout to my former students who are now taking calculus (if you don’t know any calculus, just hang tight… there will be more calculusless math goodness coming your way soon). This post is 100% moneyback … Continue reading
Posted in calculus, famous numbers, proof
46 Comments