# Tagged Questions

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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### What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$ for certain algebraic functions $a(x)$. For example, an ...
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### How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
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### Closed form solution for the zeros of an infinite sum

Does there exist a closed form expression for the zeros of the following equation? $$\sum\limits_{n=1}^\infty\frac{1}{n^4 - x^2} = 0 \text{ where } x \in \rm \mathbb R$$ Could you suggest a ...
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### How show that $a_{n}=n$ if $a_{n+1}+a_{n-1}=\frac{2n}{a_{n}-a_{n-1}}$

define sequence $\{a_{n}\}$ such $a_{1}=1,a_{2}=2$, and such $$a_{n+1}+a_{n-1}=\dfrac{2n}{a_{n}-a_{n-1}},n\ge 2$$ show that:$$a_{n}=n$$ I want use without induction solve this sequence?
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### Function that produces sequence 112123123412345…

I'm trying to find a function/formula for $a_n$ such that it produces the sequence $112123123412345$ and so on. I know that one possible way to do this is to find a function like $n-b_n$ where $b_n$ ...
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### Supremum of a sequence: $x_n = c_1 + c_2 \sum_{i=1}^{n-1} x_i$

I am dealing with a sequence $$x_n = c_1 + c_2 \sum_{i=1}^{n-1} x_i$$ where $x_1 > 0$ and $c_1, c_2 > 0$ are constants. I am wondering whether one can find another series, call it $y_n$, ...
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### Find the ratio of integrals $\int_0^1 (1\pm x^4)^{-1/2}\,dx$

How to find this ratio $$\frac{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1+x^{4}}}\mathrm{d}x}{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1-x^{4}}}\mathrm{d}x}$$ without evaluating each integral? ...
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### Recursive Sum of Previous Term and its Inverse

Can anyone help me with finding a closed form for $F_n$ where $$F_0=x_0$$ $$F_{n+1}=F_n+\frac{1}{F_n}=\frac{F_n^2+1}{F_n}$$ I could imagine this already having been done, in which case I'd appreciate ...
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### Closed form solution to an ordinary differential equaiton

How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where $C_1, C_2, C_3, C_4\in \mathbb{R}$ are all constants. It looks ...
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### I am looking for a disproof of this conjecture on closed forms

Assume $I=\int_0^\infty f(x)\text{ d}x$ and $J=\sum_{n=0}^\infty f(n) ; I,J\in\Bbb{R}$ Conjecture: If $I$ has a closed form, then $J$ must carry a closed form. Can someone find a proof or ...