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Fundamentals

Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral

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

- An integral with $2017$
- How to find the closed form of the integral$\int_{0}^{\infty}f(x)\frac{\sin^nx}{x^m}dx$
- Evaluating $\int_0^{\pi/2}({x \over \sin x})^2dx$ using value of a given integral
- Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $
- Why does $\left(\int_{-\infty}^{\infty}e^{-t^2} dt \right)^2= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy$?
- Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

(see Why do we need an integral to prove that $\frac{22}{7} > \pi$?)

and the following equivalent form of Lehmer series

$$\pi-3=\sum_{k=1}^\infty \frac{24}{(4 k+1) (4 k+2) (4 k+4)}$$

(see http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf, page 139)

These are direct proofs that $\pi>3$ and $\frac{22}{7}>\pi$, because of the positiveness of the closed form under the summation and the integral, respectively.

Symmetry

Evaluating Lehmer series leads to the integral

$$\pi-3=\int_0^1 \frac{4x^4(1-x)(2+x)}{(1+x)(1+x^2)}dx$$

Another relationship is given by the integral

$$\pi-3=\int_{0}^{1} \frac{2 x (1-x)^2}{1+x^2} dx$$

and the series

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

A series to prove $\frac{22}{7}-\pi>0$ is given by

$$\sum_{k=1}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)}=\frac{22}{7}-\pi$$

which generalizes to

$$\sum_{k=n}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)}=\int_0^1 \frac{x^{4n}(1-x)^4}{1+x^2}dx$$

With this expression approximating fractions from the integrals in the RHS may be computed sequentially by adding the next term in the series.

For $n=0$ we have

$$

\begin{align}

\frac{10}{3}-\pi

&=

\sum_{k=0}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)} \\

&=

\int_0^1 \frac{(1-x)^4}{1+x^2}dx \\

\end{align}$$

and the same difference may be obtained from a series with three factors in the denominator and its corresponding integral

$$\begin{align}

\frac{10}{3}-\pi

&=

\sum_{k=0}^\infty \frac{24}{(4k+4)(4k+6)(4k+7)} \\

&=

\int_{0}^{1} \frac{4x^3(1-x)(1+2x)}{(1+x)(1+x^2)}dx \\

\end{align}$$

More convergents

For the third convergent, we have Lucas integral and a series

$$\begin{align}

\pi-\frac{333}{106}

&=

\int_0^1 \frac{x^5(1-x)^6(197+462x^2)}{530(1+x^2)}dx \\

&=

\frac{48}{371} \sum_{k=0}^\infty \frac{118720 k^2+762311 k+1409424}{(4 k+9) (4 k+11) (4 k+13) (4 k+15) (4 k+17) (4 k+19) (4 k+21) (4 k+23)} \\

\end{align}$$

(see https://math.stackexchange.com/a/1593090/134791)

Finally, Lucas integral for the fourth convergent is

$$\frac{355}{113}-\pi=\int_{0}^{1} \frac{x^8(1-x)^8(25+816x^2)}{3164(1+x^2)}dx$$

(see http://educ.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf)

QIs there a series for $\frac{355}{113}-\pi$?

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- History Behind Integral Error Between $\pi$ and $22/7$
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I have no proof of it, but,

$\displaystyle \dfrac{355}{113}-\pi=\sum_{k=0}^{+\infty}\dfrac{1489440+10321440k}{1243(4k+13)(4k+15)(4k+17)(4k+19)(4k+21)(4k+23)(4k+25)(4k+27)}$

I got this “equality” (Wolfram Alpha confirms it) playing around with lindep function of PARI GP (LLL stuff)

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