Linked Questions

14 votes
6 answers
3k views

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 ...
user140374's user avatar
7 votes
9 answers
25k views

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 ...
pimvdb's user avatar
  • 1,283
28 votes
4 answers
1k views

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 $\mathrm{A002485}(n)/\mathrm{A002486}(n)$ ?...
Alex's user avatar
  • 76
26 votes
2 answers
740 views

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-...
clathratus's user avatar
  • 17.3k
41 votes
1 answer
1k views

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+...
Tito Piezas III's user avatar
26 votes
1 answer
1k views

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-...
LHF's user avatar
  • 8,521
7 votes
5 answers
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 ...
Ranger-of-trente-deux-glands's user avatar
1 vote
2 answers
2k views

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: (...
Jiew Meng's user avatar
  • 4,603
8 votes
3 answers
786 views

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....
RE60K's user avatar
  • 17.7k
6 votes
1 answer
155 views

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....
joriki's user avatar
  • 239k
2 votes
1 answer
857 views

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 ...
Jaume Oliver Lafont's user avatar
7 votes
1 answer
847 views

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....
Jaume Oliver Lafont's user avatar
11 votes
1 answer
337 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 ...
Jaume Oliver Lafont's user avatar
3 votes
2 answers
185 views

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 ? $$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx = \frac{22}{7} - \pi$$
Alan's user avatar
  • 2,249
5 votes
1 answer
285 views

An integral for $\frac{9801}{2206\sqrt{2}}-\pi$

From integrals $$\pi=\frac{24\sqrt{2}}{11} + \frac{8}{11} \int_0^1 \frac{x (1 - x)^2(1 + 2 \sqrt{2} x^4)}{1 + x^2 + x^4 + x^6} dx$$ and $$\pi=\frac{20\sqrt{2}}{9} - \frac{2\sqrt{2}}{3} \int_0^1 \...
Jaume Oliver Lafont's user avatar

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