# Questions tagged [pi]

The number $\pi$ is the ratio of a circle's circumference to its diameter. Understanding its various properties and computing its numerical value drove the study of much mathematics throughout history. Questions regarding this special number and its properties fit in here.

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### Does a closed-form expression exist for $\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx$?

I am trying to find a closed-form expression for the following integral $$\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx$$ There are specific values that I would like to generate (Table of ...
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### Find a simple proof that π is irrational

I know there are many questions on the site about finding a proof that π is irrational, but I'm posting the question separately to discuss a particular proof further We know that the Wallis Product is ...
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### What is the limit $\lim_{n\to \infty} \cos(n)^{n^2}$

Two years ago I found this question and wasn't able to make the slightest advancement. $\lim_{n\to \infty} \cos(n)^{(n²)}$ The question is does the limit exist, where n is natural only (of course it ...
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### Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers.

This is one of the exercises in my abstract algebra book (Nicholson) and it's just the title: Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers. All I know what to do ...
1 vote
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### How do I calculate the digit number n of pi?

I want to create a pi calculator with no limit to precision, but I cannot find a formula for finding only the digit number n of pi. Here is what I mean: Let's say I want to find the 4th digit of pi. ...
292 views

### Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
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### Approximating $\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$ [duplicate]

Consider the series $$\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$$ How many terms of this series do I need to consider to have an approximation of $\pi$ accurate up to $10$ decimal places (for ...
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### Identity involving the number pi [duplicate]

Why is this identity true? $\pi$ = $\lim_{n\to\infty} 2^n\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+...}}}}$ Where the number of two's inside the big square root is equal to n.
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### Conjecture: In Pascal's triangle with $n$ rows, the proportion of numbers less than the centre number approaches $e^{-1/\pi}$ as $n\to\infty$.

Consider Pascal's triangle with $30$ rows (the top $1$ is the $0$th row). The centre number is the number in the middle of row $30\times \frac23=20$, which is $\binom{20}{10}=184756$. The proportion ...
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### Deriving the continued fraction for Pi [closed]

So I was searching online for methods to approximate Pi and found this continued fraction that supposedly approximates to Pi when continued infinitely. I've tried searching all over the internet for ...
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1 vote
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### Is $\sqrt{\pi}=2\sum_{k=0}^{\infty} \frac{e^{-k^2}-e^{-(k+1)^2}}{2k+1}$?

Is it true that $$\sqrt{\frac{\pi}{4}}=\sum_{k=0}^{\infty} \frac{e^{-k^2}-e^{-(k+1)^2}}{2k+1}$$ Context: Attempting to find an easier proof for this estimate, when $x=1$. My attempt: Leibniz's ...
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### $\pi$ approximation method confusion

I am reading a book (A History of Pi) in it there is a story about how Indian mathematicians found the value of $\pi$ by inscribing the polygons in a circle with diameter of 100 and doubling the sides ...
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### A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$

I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
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### Is there a known explanation for the Feynman point?

The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, ...
1 vote
I wanted to know if the method I used is valid and not just pure luck: Consider the function: $$\dfrac{\cos(\frac{\pi x}{2})}{(x^2-1)}$$ The roots of the equation are$\quad\pm3,5,7,9,\cdots$ So the ... 