Linked Questions
21 questions linked to/from Is there an integral that proves $\pi > 333/106$?
14
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How do I prove that $3<\pi<4$?
Let's not invoke the polynomial expansion of $\arctan$ function.
I remember I saw somewhere here a very simple proof showing that $3<\pi<4$ but I don't remember where I saw it.. (I remember ...
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7
votes
9
answers
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Calculate $\pi$ precisely using integrals?
This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$.
We would get the surface ...
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28
votes
4
answers
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Seeking proof for the formula relating Pi with its co seenvergents
Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via $\mathrm{A002485}(n)/\mathrm{A002486}(n)$ ?...
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26
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A magnificent series for $\pi-333/106$
Stated here without proof is the magnificent series $$\frac{48}{371} \sum_{k=0}^\infty \frac{118720 k^2+762311 k+1409424}{(4 k+9) (4 k+11) (4 k+13) (4 k+15) (4 k+17) (4 k+19) (4 k+21) (4 k+23)} \\=\pi-...
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41
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answer
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Is there an integral for $\pi^4-\frac{2143}{22}$?
In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$),
$$\pi^4 \approx 2^4+...
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26
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answer
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Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$
Prove that:
$$e^\pi+\frac{1}{\pi}< \pi^{e}+1$$
Using Wolfram Alpha $\pi e^{\pi}+1 \approx 73.698\ldots$ and $\pi(\pi^{e}+1) \approx 73.699\ldots$
Can this inequality be proven without brute-...
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5
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Showing $\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$ without a calculator
Show that:
$$\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$$
I have tried power series of exponential around $0$ wich is :
$$e^x=1+x+\frac{x^2}{2}+O(x^3)$$
We can ...
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1
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2
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Finding $\int_0^1{\frac{x^4(1-x)^4}{1+x^2}}dx$
The question I am working on: Evaluate
$$\frac{1}{2} \int^1_0{x^4 (1-x)^4 } dx \le \int^1_0{\frac{x^4 (1-x)^4}{1+x^2}} dx \le \int^1_0{x^4 (1-x)^4 } dx$$
So using integration by parts to solve:
(...
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8
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3
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Calculating value of $\pi$ independently using integrals.
Recently I noticed this integral:
$$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}7-\pi\approx0$$
Which is a very interesting result which gives us the value of $\pi\approx\frac{22}7\approx3....
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6
votes
1
answer
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Succinct proof that $\frac\pi4+\frac\pi6+\log2\gt2$
In answering Average angle between two randomly chosen vectors in a unit square, I noticed that the average angle formed by two vectors uniformly picked in the unit square, $\frac\pi4+\log2-1\approx0....
- 230k
2
votes
1
answer
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Series and integrals for inequalities and approximations to $\pi$
Fundamentals
Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral
$$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$
(see Why do we need an integral to prove ...
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votes
1
answer
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Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+\log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?
The eighth harmonic number happens to be close to $e$.
$$e\approx2.71(8)$$
$$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$
This leads to the almost-integer
$$\frac{e}{H_8}\approx1....
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votes
1
answer
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Asymptotic quality of rational approximations to $\pi$
Dalzell's integral
$$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$
is case $n=2$ of the generalization
$$\int_0^1 \frac{x^{n+2}(1-x)^{2n}}{2^{n-2}(1+x^2)}dx = \frac{p_n}{q_n}-\pi$$
Such an ...
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2
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A series of positive terms to prove $\pi>\frac{333}{106}$
This is a consequence of the answer to that question.
A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms
$$\pi-\frac{333}{106} \\
=\frac{48}{371} \sum_{k=0}^\infty \frac{...
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A Definite Integral Whose Value Will Be Familiar To Everyone?
Here's a definite integral whose value carries memories of grade school. Is there a useful generalization ?
$$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx = \frac{22}{7} - \pi$$
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