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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 convergents

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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540 views

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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