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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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$\frac{e}{H_8}\approx1.... • 5,174 11 votes 1 answer 331 views ### Asymptotic quality of rational approximations to \pi Dalzell's integral$$\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$$Such an ... • 5,174 -2 votes 2 answers 265 views ### A series of positive terms to prove \pi>\frac{333}{106} This is a consequence of the answer to that question. A proof that \pi > \frac{333}{106} is given by the series of positive terms$$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty \frac{...
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Here's a definite integral whose value carries memories of grade school. Is there a useful generalization ? $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx = \frac{22}{7} - \pi$$