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 the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau$ is a constant. But is ...
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Analytic extension of $\sum_{k=1}^n\frac1k$ complex domain

The analytic extension: $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I was wondering for what values of $n$ does this extend to, mainly complex values of $n$. I know it is defined for $n=0$, ...
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Evaluating the integral $\int \frac{x^2+x}{(e^x+x+1)^2}dx$

Evaluate $$\int \frac{x^2+x}{(e^x+x+1)^2}dx$$ I tried converting in the form of Quotient rule(seeing the square in the denominator), neither am I able to make the denominators' derivative in the ...
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Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
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Deducing the closed form for pentagonal numbers

Consider the sequence: $0,1,5,12,22,35,51,70,92,117,145,176,\ldots$ Find both a recurrence and a closed form for this sequence. I've done some research and found out that the majority of ...
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Proving $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - …= 1 + \frac{1}{2} +…+ \frac{1}{n}$ [closed]

Prove that $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - ... (-1)^{n+1}\frac{_{n}^{n}\textrm{C}}{n} = 1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n}$ I am not able to prove this....
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Finding a closed form for $\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$

I am trying to find a closed form for $S=\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$, $k \in \mathbb{N^{*}}$, $a>0$ I don't even bother to look for a closed form with an odd exponent, ...
Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?
Nested squares seem to be more promising than nested radicals, since they give rational approximations and in principle can be expanded into a series. These two expressions converge numerically: \...
We know the closed form for Fibonacci number as $F_n=\frac{1}{\sqrt5}\left[\left(1+\frac{\sqrt5}{2}\right)^n−\left(1−\frac{\sqrt5}{2}\right)^n\right]$ But while finding $F_n \pmod{99991}$ the closed ...