Simplest way to get the lower bound $\pi > 3.14$

Question

Inspired from this answer and my comment to it, I seek alternative ways to establish $\pi>3.14$. The goal is to achieve simpler/easy to understand approaches as well as to minimize the calculations involved. The method in my comment is based on Ramanujan's series $$\frac{4}{\pi}=\frac{1123}{882}-\frac{22583}{882^{3}}\cdot\frac{1}{2}\cdot\frac{1\cdot 3}{4^{2}}+\frac{44043}{882^{5}}\cdot\frac{1\cdot 3}{2\cdot 4}\cdot\frac{1\cdot 3\cdot 5\cdot 7}{4^{2}\cdot 8^{2}}-\dots\tag{1}$$ This is quite hard to understand (at least in my opinion, see the blog posts to convince yourself) but achieves the goal of minimal calculations with evaluation of just the first term being necessary.

On the other end of spectrum is the reasonably easy to understand series $$\frac\pi4=1-\frac13+\frac15-\cdots\tag2$$ But this requires a large number of terms to get any reasonable accuracy. I would like a happy compromise between the two and approaches based on other ideas apart from series are also welcome.

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A previous question of mine gives an approach to estimate the error in truncating the Leibniz series $(2)$ and it gives bounds for $\pi$ with very little amount of calculation. However it requires the use of continued fractions and proving the desired continued fraction does require some effort.

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Another set of approximations to $\pi$ from below are obtained using Ramanujan's class invariant $g_n$ namely $$\pi\approx\frac{24}{\sqrt{n}}\log(2^{1/4}g_n)\tag{3}$$ and $n=10$ gives the approximation $\pi\approx 3.14122$ but this approach has a story similar to that of equation $(1)$.

Answer 1

From the elementary inequality $$\frac{\sin x}x\le\frac{2+\cos x}3,$$ we get with $x=\pi/6$ easily $\pi\ge\frac{18}{4+\sqrt{3}}=3.1402\ldots$
Proof of the inequality (elementary, though not obvious): let $$f(x)=\frac{\sin x}{x(2+\cos x)}.$$ In order to prove $f(x)\le\lim_{x\to+0}f(x)$, we prove $f(x)\le f(x/2)$. That follows from $$f(x)=f(x/2)\,\frac{(2+\cos x/2)\cos x/2}{1+2\cos^2 x/2},$$ since with $c=\cos x/2$, we have $$\frac{(2+c)c}{1+2c^2}=1-\frac{(1-c)^2}{1+2c^2}\le1.$$

Answer 2

Take Machin's formula: $$\pi=16\tan^{-1}\frac15-4\tan^{-1}\frac1{239}$$ Expand the arctangents into their Taylor series: $$\pi=16\left(\color{blue}{\frac15-\frac1{5^3×3}}+\frac1{5^5×5}-\dots\right)-4\left(\color{blue}{\frac1{239}}-\frac1{239^3×3}+\dots\right)$$ $$=\color{blue}{\frac{16}5-\frac{16}{375}-\frac4{239}}+\delta$$ $$=\color{blue}{3.140596\dots}+\delta$$ with $0<\delta<\frac{16}{5^5×5}+\frac4{239^3×3}$ since the two series are alternating with term magnitudes strictly decreasing. This proves $\pi>3.140596\dots>3.14$.

Answer 3

If we consider the Beuker-like integral $$ 0<\int_{0}^{1}\frac{x^8(1-x)^8}{1+x^2}\,dx = 4\pi-\frac{188684}{15015} $$ we get, through partial fraction decomposition and few operations in $\mathbb{Q}$, $$ \pi > \frac{47171}{15015} > 3.14159.$$

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Inspired by Professor Vector's brilliant approach, I am adding a further approach.
By the Shafer-Fink inequality we have $\arctan(x)>\frac{3x}{1+2\sqrt{1+x^2}}$ for any $x>0$, hence by evaluating both sides at $x=\frac{1}{\sqrt{3}}$ we get $\pi>\frac{18}{13}(4-\sqrt{3})=3.140237\ldots$ A refinement of the previous inequality is $$\forall x>0,\qquad \arctan(x)>\frac{6x}{1+\sqrt{1+x^2}+2\sqrt{2}\sqrt{1+x^2+\sqrt{1+x^2}}} $$ and the evaluation at $x=\frac{1}{\sqrt{3}}$ produces the sharper bound $$ \pi > \frac{36}{2+\sqrt{3}+4 \sqrt{2+\sqrt{3}}} > 3.1415.$$

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Yet another approach. The inequality $\arctan(x)>\frac{5x(21+11x^2)}{105+90x^2+9x^4}$ for any $x\in(0,1)$ comes from the Gauss continued fraction / the Padé approximants for the arctangent function. By replacing $x$ with $\frac{x}{1+\sqrt{1+x^2}}$, then evaluating at $x=\frac{1}{\sqrt{3}}$, we get the nice and tight approximation: $$ \pi > \color{blue}{\frac{5}{601}\left(944-327\sqrt{3}\right)}>3.141592.$$

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There is also a nice geometric argument leading to an acceleration of Vieta's formula. Let $PQ$ be a side of a regular $n$-agon inscribed in a unit circle centered at $O$. Let $M$ be the midpoint of the minor arc $PQ$. We may consider the unique parabola through $P,M,Q$ and approximate the area of the circle sector delimited by $P,O,Q$ through $[POQ]$ plus the area of a parabolic segment, $\frac{4}{3}[PMQ]:$

It follows that if $A_n$ is the area of the inscribed $2^n$-agon, we have $$ \pi \geq A_n+\frac{4}{3}\left(A_{n+1}-A_n\right) = \frac{4}{3}A_{n+1}-\frac{1}{3}A_n $$ with $$ A_n = 2^{n-1} \sin\frac{\pi}{2^{n-1}} $$ and $\{A_n\}_{n\geq 2}$ being computable through a simple recursion, $$ A_{n+1} = 2^n \sqrt{\frac{1-\sqrt{1-\frac{A_n^2}{4^{2n-2}}}}{2}}$$ where $$ \frac{4}{3}A_5-\frac{1}{3}A_4 = \frac{4}{3}\left[8\sqrt{2-\sqrt{2+\sqrt{2}}}-\sqrt{2-\sqrt{2}}\right]=\color{green}{3.141}44\ldots $$ By considering a $12$-agon and a $24$-agon we get the simpler $$ \pi > 4\sqrt{2}(\sqrt{3}-1)-1 = \color{green}{3.141}10\ldots $$

Answer 4

Similarly to Jack d'Aurizio's answer (first section), we have the simpler integral

$$\frac{1}{2}\int_0^1 \ \frac{x^3(1-x)^6}{1+x^2} dx = \pi-\frac{1759}{560}=\pi-\left(3.14+\frac{3}{2800}\right)$$

A similar one evaluating to exactly $$\pi-3.14=\pi-\frac{157}{50}$$ can be obtained with the methods of Lucas (http://educ.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf)

Since

$$3.14=\frac{157}{50}=\frac{22}{7}-\frac{1}{350},$$

the result can be obtained subtracting Dalzell's integral

$$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx =\frac{22}{7}-\pi$$

from $$\frac{4}{5}\int_0^1 x^3(1-x)^4 dx = \frac{1}{350},$$

which gives

$$\frac{1}{5}\int_0^1 \frac{x^3(1-x)^4(4-5x+4x^2)}{1+x^2} dx = \pi -\frac{157}{50} > 0 $$

This is a direct proof for $\pi>\frac{157}{50}$ because the integrand is non-negative in $(0,1)$, very close to this proof that $\frac{22}{7}$ exceeds $\pi$.

The corresponding series is

$$1152\sum_{k=0}^\infty \frac{2k^2+8k+13}{(4k+4)(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)(4k+12)}=\pi-\frac{157}{50}$$

(check

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We can also take the first term out of a series relating $\pi$ with the approximation from below $\frac{25}{8}$ to get

$$\begin{align} \pi &= \frac{25}{8} + \sum_{k=0}^\infty \frac{360}{(4k+2)(4k+4)(4k+5)(4k+7)(4k+8)(4k+10)}\\ &=\frac{1759}{560} + \sum_{k=1}^\infty \frac{360}{(4k+2)(4k+4)(4k+5)(4k+7)(4k+8)(4k+10)}\\ \end{align}$$

where $\frac{1759}{560}>3.14$ as shown above.

This fraction already appeared in the first integral, but proving the series as in this answer now gives

$$\frac{1}{4}\int_0^1 \frac{x^5(1-x)^4(1+4x+x^2)}{1+x^2}dx = \pi -\frac{1759}{560}$$

Answer 5

Another simple method is to use the Newton-Raphson method to find the zero of the function $\sin(x)$ at $x = \pi$ using the initial guess of $x = 3$. The fact that at $x = \pi$ the function $\sin(x)$ has a point of inflexion, means that convergence to $x = \pi$ will happen in an alternating way, therefore you obtain rigorous upper and lower bounds. So, the sequence defined by the recurrence:

$$x_{n+1} = x_n - \tan(x_n)$$

and $x_0 = 3$ converges to $\pi$. We have $x_1 = 3.1425\cdots$, $x_2 = 3.1415926533\cdots$, and therefore $\pi > 3.1415926533$. Now, we need to compute $x_2$ here without using $\pi$ itself, we can use the series expansions of $\sin(x)$ and $\cos(x)$ around $x = 0$. These are both alternating series, so we can easily get to an accurate bound on $x_1$ and from that we can get to an accurate bound on $x_2$, the lower bound of which then yields an accurate lower bound for $\pi$.

Answer 6

For modest requirements of accuracy, we can use a quite basic and easy to understand convergence acceleration technique to transform the Gregory/Leibniz series into something that yields the result with little computation.

If we have an alternating series

$$\sum_{n = 0}^{\infty} (-1)^n a_n$$

where the $a_n$ are slowly converging to $0$, then it takes little guessing to believe that the arithmetic mean of two successive partial sums of the series is a much better approximation to the value of the series than either or the two partial sums. We can write the mean as

$$\sum_{n = 0}^{m-1} (-1)^n a_n + \frac{(-1)^m a_m}{2} = \frac{a_0}{2} + \frac{1}{2} \sum_{n = 0}^{m-1} (-1)^n(a_n - a_{n+1})$$

and see that if $\bigl(a_n - a_{n+1}\bigr)_{n\in \mathbb{N}}$ is a decreasing sequence we immediately have an error bound of $\frac{1}{2}(a_{m} - a_{m+1})$ for this arithmetic mean. If $(a_n)$ is slowly converging, then this is much smaller than either of $a_m$ and $a_{m+1}$. For nice $(a_n)$, this can be iterated.

If we apply that to $a_n = \frac{1}{2n+1}$, we find $a_n - a_{n+1} = \frac{2}{(2n+1)(2n+3)}$, which is decreasing - and still converges slowly to $0$, so we get a significant improvement from then applying the method again. We get

\begin{align} \frac{\pi}{4} &= \sum_{n = 0}^{\infty} \frac{(-1)^n}{2n+1} \\ &= \frac{1}{2} + \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)} \\ &= \frac{1}{2} + \frac{1}{6} + 2\sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)(2n+5)} \\ &= \frac{1}{2} + \frac{1}{6} + \frac{1}{15} + 6\sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)(2n+5)(2n+7)} \end{align}

and so on, each series converging faster than the previous. Since the terms get more complicated, it is computationally - at least when done by hand - better to start accelerating not right at the beginning of the series, but rather compute a partial sum of the original series, and then use acceleration only on the remainder. Let's say that starting the remainder at $n = 4$ wouldn't need an unreasonable amount of computation, then we get

\begin{align} \frac{\pi}{4} &= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \sum_{n = 4}^{\infty} \frac{(-1)^n}{2n+1} \\ &= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{18} + \sum_{n = 4}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)} \\ &= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{18} + \frac{1}{198} + 2\sum_{n = 4}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)(2n+5)} \\ &= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{18} + \frac{1}{198} + \frac{1}{1287} + 6\sum_{n = 4}^{\infty} \frac{(-1)^n}{(2n+1)(2n+3)(2n+5)(2n+7)} \\ &> 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{18} + \frac{1}{198} + \frac{1}{1287} \\ &> 0.78519 \end{align}

and thus $\pi > 3.14076$.

Of course Machin-like formulæ are far more efficient. But it's not too terrible.

Answer 7

A very easily understandable way is to use regular polygons inscribed into a circle of radius $1$. If $a_n$ is the length of the side of the $n$-gon, then $$a^2_{2n} = 2-\sqrt{4-a^2_n}\text{.}$$  A lower bound on $a_n$ implies a lower bound on $a_{2n}$, we can thus round down intermediary results. Four applications of the formula with 4 to 5 significant digits are enought to establish the bound of $3.14$. $$\begin{array}{rl} a^2_6 =& 1\\ a^2_{12} >& 0.2679 \\ a^2_{24} >& 0.06813 \\ a^2_{48} >& 0.017105 \\ a^2_{96} >& 0.004280 \\ \end{array}$$ Leading to $a_{96} > 0.06542$ and $\pi > 48a_{96} > 3.14$.

Answer 8

Since $\cos \frac\pi3 = \frac12$ and the cosine is strictly decreasing between $0$ and $\pi$, we can prove that $3.14 < \pi$ by showing that $\cos\frac{3.14}3 > \frac12$. This takes only 4 terms of the power series for cosine:

x^0/0! = 1   adding gives 1
x^2/2! = 0.547755555555556   subtracting gives 0.452244444444444
x^4/4! = 0.0500060247736625   adding gives 0.502250469218107
x^6/6! = 0.00182607185873483   subtracting gives 0.500424397359372

Since the series is alternating and absolutely decreasing, when we have just subtracted a term and gotten something above $\frac12$ we now know that the limit is larger than $\frac12$.

Edit: showing that $\sin\frac{3.14}6 < \frac12$ is slightly faster:

x^1/1! = 0.523333333333333   adding gives 0.523333333333333
x^3/3! = 0.0238882283950617   subtracting gives 0.499445104938272
x^5/5! = 0.000327122745394376   adding gives 0.499772227683666

One can do this with pen and paper, keeping 4 digits after the decimal point in every intermediate (and explicit interval arithmetic), using just two 4-by-4 multiplications and two 4-by-3 to show $3.141<\pi$:

Five terms to 10 decimal digits will show that $3.14159 < \pi < 3.14160$.

Answer 9

This is a slight variant on Henning Makholm's answer, with an emphasis on actually carrying out the computations that are required to verify the desired inequality.

Let $u=.314$. It suffices to show that $\sin u\lt\sin\left({\pi\over10}\right)={\sqrt5-1\over4}$. But

$$\sin u\lt u-{u^3\over6}+{u^5\over120}\lt u\left(1-{u^2\over6}+\left({u^2\over6}\right)^2\right)$$

At this point it's not too hard to get a calculator to tell you that

$$.314\left(1-{.314^2\over6}+\left(.314^2\over6\right)^2\right)\approx0.3089249\lt.309017\approx{\sqrt5-1\over4}$$

but let's do it showing (almost) all the arithmetic to establish the inequalities

$$.314\left(1-{.314^2\over6}+\left(.314^2\over6\right)^2\right)\lt.309\lt{\sqrt5-1\over4}$$

The second inequality is equivalent to $2.236\lt\sqrt5$. Since $2236/4=559$, this is equivalent to $559^2\lt5\cdot250^2=312500$. It's not too hard (especially if you use a calculator...) to compute $559^2=312481$, which verifies the inequality. (Note, I consider $5\cdot25^2=5\cdot625=3125$ to be an "easy" multiplication.)

As for the first inequality, since $.309=.314-.005$, it is equivalent to

$${.314^2\over6}-\left(.314^2\over6\right)^2\gt{5\over314}$$

The (not too hard) computation $314^2=98596$ followed by division by $6$ shows that

$$.0164\lt{.314^2\over6}\lt.02$$

It follows that

$${.314^2\over6}-\left(.314^2\over6\right)^2\gt.0164-.0004=.016={16\over1000}={2\over125}\gt{5\over314}$$

where the final inequality is verified by $2\cdot314=628\gt625=5\cdot125$.

If there is an easier way to organize the arithmetic so that you don't have to do a couple of messy multiplications (e.g., $559\times559$ and $314\times314$) somewhere along the line, I'd like to see it. (Even Professor Vector's marvelous answer, when ${18\over4+\sqrt3}\gt3.14={157\over50}$ is simplified to the inequality $900-628=272\gt157\sqrt3$, seems to require computing $272^2$ and $157^2$.)

Answer 10

From $$\pi=\sum_{k=0}^{+ \infty}{\dfrac {1}{16^k}} \cdot \dfrac {120k^2+151k+47}{512k^4+1024k^3+712k^2+194k+15}$$ you obtain your inequality by summing just the first two terms.

Answer 11

Do you want a solution that is a compromise between calculation efficiency and ease of understanding? This story might help.

Many, many years ago in school, I was introduced to Fortran and had limited access to a computer. It was physically large but very low power by today's standards. I knew the famous $\frac{\pi}{4} = \tan^{-1}(1)$ formula but I also realised how slowly it converges. I would not be able to get far with the run time available to me.

The internet did not exist yet and the school's and the local public library did not help. I knew how the $\frac{\pi}{4}$ was derived so I played with other trig formulae. I managed to calculate the Taylor series of $\sin^{-1}$ by a mixture of messy differentiation and induction. I figured that if I evaluated $\frac{\pi}{6} = \sin^{-1}(\frac{1}{2})$, the convergence would be linear: twice the terms would give me twice the number of decimal places. In practice, the algorithm was quadratic since if I aimed at $n$ times as many decimal places, I would need $n$ times as many terms and each calculation would take $n$ as long.

I forget the run time that I had but the best I achieved was 500 decimal places.

Many years later but still long ago, I rewrote the program in C and ran it on an idle Unix system at work. In a month, it calculated a million decimal places. For comparison, I ran it on my laptop a few years ago, a million places took 3.5 hours. My Raspberry Pi required 44 hours.

Answer 12

Equation (2) in the question is the result of plugging $x=1$ in the expansion for the arctangent, using

$$\frac{\pi}{4}=\tan^{-1}(1)$$

Instead, from $$\frac{\pi}{6}=\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)$$ a series with faster convergence is obtained. Taking six terms,

$$\pi>2\sqrt{3}\left(1-\frac{1}{3·3}+\frac{1}{5·3^2}-\frac{1}{7·3^3}+\frac{1}{9·3^4}-\frac{1}{11·3^5}\right)=\frac{509024\sqrt{3}}{280665}>3.141$$

Four terms from $$\frac{\pi}{8}=\tan^{-1}\left(\sqrt{2}-1\right),$$ give

$$\pi> 8\left(\sqrt{2}-1-\frac{(\sqrt{2}-1)^3}{3}+\frac{(\sqrt{2}-1)^5}{5}-\frac{(\sqrt{2}-1)^7}{7}\right)=\frac{32}{105}\left(716-499\sqrt{2}\right)>3.141,$$

while two terms from $$ \frac{\pi}{12}=\sin^{-1}\left( \frac{\sqrt{3}-1}{2\sqrt{2}} \right)$$

lead to $$\pi > 12\left( \frac{\sqrt{3}-1}{2\sqrt{2}} + \frac{1}{6} \left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)^3 \right) = \frac{27\sqrt{3}-29}{4\sqrt{2}}>3.14$$

Answer 13

From series $$\pi^6=960\sum_{k=0}^\infty \frac{1}{(2k+1)^6},$$

$$\pi^6>960$$

But $3.14^6<960$, so $3.14^6<\pi^6$ and therefore $3.14<\pi$.