4

The geometrical proof is simple, but I want to know how to prove $\pi \gt 3$ by using basic calculus?

Jaume Oliver Lafont
  • 5,212
tmpbin
  • 227
  • 4
    How do you have $\pi$ defined? If you have it defined as or can use the result $\pi = 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\dots)$ you will see after the seventh term the partial sums will remain above $3$ – JMoravitz Jun 17 '17 at 16:30
  • 4
    $\sin(y) < y$ for $y>0$, let $y=\pi /6$ – Spine Feast Jun 17 '17 at 16:32
  • Note the following post, using geometry: Prove that $\pi > 3$ – amWhy Jun 17 '17 at 16:39
  • 1
    @amWhy Not a duplicate of any of the linked questions. Also, this question got some good and interesting answers, much better than the linked questions. Please consider to reopen. – WimC Jun 18 '17 at 04:46

4 Answers4

6

$$\sin(x)<x$$ whenever $x>0$. Take $x=\pi/6$.

ADDED IN EDIT

Apologies to Spine Feast who just pipped me to this, but $$\frac\pi6=\int_0^{1/2}\frac{dt}{\sqrt{1-t^2}}>\int_0^{1/2}dt=\frac12$$ is the same thing again but even more analytical.

Angina Seng
  • 158,341
6

The integral

$$\int_0^1 \frac{x(1-x)^2}{1+x^2}dx$$

evaluates to $\frac{\pi-3}{2}$ and the result is positive because the integrand is $\ge0$ in the interval $(0,1)$.

A related series is $$\pi = 3+ \sum_{k=0}^\infty \frac{24}{(4k+2)(4k+3)(4k+5)(4k+6)}$$

Jaume Oliver Lafont
  • 5,212
2

There is a feature column in AMS titled Calculating Pi using Elementary Calculus that you might want to check out.

Jose Arnaldo Bebita Dris
  • 1
1

I'm sure there are many, but here is a proof that uses the famous infinite sum

$$ \sum_{n=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6} $$ Solving for $\pi$ and using the fact that an infinite sum of positive terms increases monotonically to its limit, we see that

$$ \pi = \left(6\sum_{n=1}^\infty\frac{1}{n^2}\right)^{1/2} > \left(6\sum_{n=1}^7\frac{1}{n^2}\right)^{1/2}> \sqrt{9} = 3 $$ You can see that $6\sum_{n=1}^7 \frac{1}{n^2} > 9$ by direct calculation.

icurays1
  • 17,161