# Tagged Questions

For questions about approximating real numbers by rational numbers.

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### Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
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### A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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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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### 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{...