# Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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### How do I prove that an equation is bounded?

So I have a budget set that is B= {(x,y) |x ≥0,y ≥0,px(x) + py(y) ≤M}. How do i prove that is a closed and bounded set ? I'm trying to maximise the utility function U (x,y) by choosing the optimal ...
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### Does $\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdot\cdot\cdot\right)^{\frac{1}{x}}=L$ admits a closed form?

I try to simplify this limit : $$\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdots\right)^{\frac{1}{x}}=L$$ Where we compose the Gamma function with itself . From the past ...
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### How to determine whether a given number is a member of given by the that recurrence sequence

Given recurrent sequence: $x_1$; $x_2 = (x_1^4 + 126x_1^2 - 1323)/8x_1^3$; $x_3 = (x_2^4 + 126x_2^2 - 1323)/8x_2^3$; ... $x_n = ( x_{n-1}^4 + 126x_{n-1}^2 - 1323)/8x_{n-1}^3$; ... How to determine ...
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### Closed form for an alternating values infinite sequence of nested roots

I tried to derive a closed form for an infinite sequence of nested square roots with alternating values and found myself with a 4th degree equation, which I'm fine with, but I was wondering if there's ...
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### Is there a general form of a logical formula with N variables?

Let N = 2. Then there are 16 possible non-equivalent N variable logical formulas, listed below. False, A ∧ B, ¬(A → B), A, ¬(B → A), B, A ⊕ B, A v B, ¬(A v B), ¬(A ⊕ B), ¬B, B → A, ¬A, A → B, ¬(A ∧ B),...
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### $\sin(\frac{\pi}{p})$ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$ ??

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
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### Is there a closed form of $\sum\limits_{r=0}^n \binom{n-r}{r}x^r$? [duplicate]

$\sum\limits_{r=0}^n \binom{n-r}{r}x^r=\sum\limits_{r=0}^{\lfloor n/2\rfloor} \binom{n-r}{r}x^r$ I need its closed form for a probability problem. I know about the case where $x=1$. It's the sum of ...
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### True or false: For continuous random variable $X$, if $E(X)$ has a closed form, then $P(X<E(X))$ has a closed form.

For this question, let us define "closed form" as an expression restricted to addition, subtraction, multiplication, and division; exponents and logarithms, including $e^x$ and $\ln{x}$; ...
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### A certain sum of multinomial coefficients

I would like to know if there is a nice expression for the sum $$S(n)=\sum_{i+j=n}\binom{3i}{i,i,i}\binom{3j}{j,j,j}$$ where $n$ is a non-negative integer. I have entered in the first few values of ...
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### What's the average dimension of the Cantor sets?

Consider the open set $E=\{(x,y)\in(0,1)^2:x+y<1\}$ and define $\text{Cant}:E\to(0,1)$ such that $x^{\text{Cant}(x,y)}+y^{\text{Cant}(x,y)}=1$ which we may call the Cantor dimension function. My ...
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### Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$
Assume $c$ is a small real number. QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$? $$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$ I got as far as ...