A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$
This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$.I. $\pi$ and $\pi^2$ There are series with all terms positive for $\pi-3$ and...
View ArticleA pattern for BBP-type pi formulas? One for order 28?
The original BBP pi formula is$$\pi = \sum_{n=0}^\infty \frac1{2^{4n}}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)$$Others are:Order 4. Let $P_k = 4n+k$$$\pi =...
View ArticleApproximating $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$ with stable...
Consider the Leibniz formula for $\pi$$$\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}.$$What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places, in the sense that...
View ArticleHow to prove this general form of the BBP formula?
QuestionHow to prove the general form of the BBP(Bailey–Borwein–Plouffe) formula?Prove this formula is always true, or find a counterexample $k$ that makes the formula invalid:$$π≟\sum_{n=0}^∞...
View ArticleFind the coincidence series with value $π-3.1$.
I discovered an interesting series that seems to yield rational numbers related to π:$$\sum_{k=0}^∞\frac{m}{\Pi_{i∈I}(4k+i)}=\left|\frac{p}{q}-π\right|$$I don't know if it's a coincidence or if the...
View ArticleExpected value of index of an n-digit number found in pi
Let $E(n)$ be the expected value of the index of finding an n-digit number in the digits of pie.g.Number 0 is found at index 32Number 1 is found at index 1Number 2 is found at index 6Number 3 is found...
View ArticleCan we find the exact value of a double sum with cosine without differentiation?
After finding an interesting double sum$$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{(-1)^{m+n}}{(m+n)^2} = \frac{\pi^2}{12}-\ln 2 ,$$I started to investigate a harder one$$\displaystyle...
View Article(1/2)! from the infinite product definition of gamma
I wanted to derive$$\left(\frac{1}{2}\right)! = \frac{\sqrt\pi}{2}$$from the infinite product definition of the gamma...
View ArticleDoubt in Niven's proof given in wikipedia.
I am writing to seek clarification on a specific aspect of Niven's proof, as presented in the Wikipedia article. I have attached an image for your reference.My inquiry pertains to the value of and its...
View ArticleFormula for Natural logarithm of $\pi$
Does any formula or expansion exist that gives $\ln \pi$ ?The expansion should not just be any formula of $\pi$ with a $\ln$ before it. For example $\ln \pi$ = k + $\sum f(x)$ or something of this type.
View ArticleDoes $\pi$ contain all possible number combinations?
$\pi$ PiPi is an infinite, nonrepeating $($sic$)$ decimal - meaning thatevery possible number combination exists somewhere in pi. Convertedinto ASCII text, somewhere in that infinite string of digits...
View ArticleCould PI have a different value in a different universe?
The value of pi is determined by the circumference of a circle.Why is it any particular constant number? Would a circle as defined as a perfect circle in any universe lead to a different value of...
View ArticleIntuitive reason for why the Gaussian integral converges to the square root...
This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an...
View Articleln n and pi breakup sequence
To compute the logarithm of an integer efficiently using pen and paper, one can first write the number as a product of numbers close to $1$. For example, $$10 =...
View ArticleUnexpected appearances of $\pi^2 /~6$.
"The number $\frac 16 \pi^2$ turns up surprisingly often and frequently in unexpected places." - Julian Havil, Gamma: Exploring Euler's Constant.It is well-known, especially in 'pop math,'...
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Conjecture: In Pascal's triangle with $n$ rows, the proportion of numbers...
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 of...
View ArticleInteresting and unexpected applications of $\pi$
$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} =...
View ArticleCould we calculate pi using an iterative series
I know that, as a hobbyist mathematician, this is generally a term we can use to express pi\begin{equation*} \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} +...
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A nice formula for pi [closed]
I would like to share a simple derivation of an iterative formula for Pi.The formula I have derived:$\begin{align}A_1 &= 4 \\A_{n+1} &=2*4^{n} \left(1 - \sqrt{1 - \frac{A_n}{4^{n}}}\right)....
View ArticlePower Series with digits of $\pi$
Sorry if this has already been asked, but I haven't found a post.Can anything be said about the function$$f(z)=\sum_{n=0}^\infty a_n z^n$$where the $a_n$'s are the digits of $\pi= 3.14159...$, so...
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