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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26
votes
0answers
441 views

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: ...
3
votes
1answer
81 views

Closed form for Fibonacci numbers

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 ...
11
votes
2answers
179 views

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 ...
1
vote
1answer
34 views

how do I prove this by induction? (recursion)

The terms are given recursively: $P_0=3$ $P_1=7$ and $P_n = 3P_{n-1}-2P_{n-2}$ for $n\ge2$ What should I assume and what step proves that $P_n=2^{n+2}-1$ is a closed form of the sequence. Suppose ...
-1
votes
2answers
59 views

How can I prove by induction that this is a closed form of the Fibonacci sequence? [duplicate]

How can I prove by induction that this is a closed form of the Fibonacci sequence? $$F_n=\frac1{\sqrt5}\left(\frac{1+\sqrt5}2\right)^{n+1}-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^{n+1}$$ I've ...
1
vote
2answers
68 views

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

I need to find a function $f(x)$ which is equal to the sum $$ \sum_{n=0}^\infty \frac{n}{n+1}x^n, $$ for every $x\in \mathbb{R}$ for which the series converge. Now, using WolframAlpha, I've found the ...
0
votes
1answer
25 views

Infinite sum over Gamma functions?

I am having quite a bit of trouble understanding this sum. Can someone explain to me exactly how to this from 1 to 3,very easily way? Question its from this webpage Thanks.
2
votes
3answers
81 views

Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$

I was working on a physics problem, where I encountered the following summation problem: $$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant ...
5
votes
1answer
107 views

Definite Integral $\int_0^1 \left \{\frac{(-1)^{\lfloor 1/x \rfloor}}{x} \right\}\, dx$

The curly brackets mean 'FractionalPart' which, I believe, is defined as {${x}$}$=x-\lfloor x \rfloor$ where $x \in \mathbb{R}$. The conjecture I have found but can not prove is that the definite ...
0
votes
0answers
40 views

Closed form for an integral involving an inverse square root and a cube

I bumped into a curious integral, which has so far resisted all the attempts I made to find a closed-form solution (not that I am sure there is one to start with). Has anyone encountered a similar ...
0
votes
2answers
137 views

Closed Form of $ \int_{0}^{\pi} x^3\ln^8(2\sin{(x)}) dx$

I have need of a closed form expression to this definite integral: $$ \int\limits_{0}^{\pi} x^3\ln^8(2\sin{(x)}) dx$$ It does exist apparently. Both limits are problematic for the natural ...
2
votes
2answers
70 views

Integral $\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left({x} \right)\,dx$

Is there a closed form for this integral? $\displaystyle \int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left({x} \right)\,dx\\$ All I have been able to find, so far, is a numeric approximation of $-1.13348$
2
votes
2answers
53 views

Is $\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$ right?

How to prove $$\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$$ I met this function when I tried to give another proof of the known lower bound of Tur\'an functions of complete ...
0
votes
0answers
49 views

Closed form for $\int_{0}^{t}\operatorname{Ai}(-z)^4\, \mathrm{d}z $

I am hoping for a closed form solution for the following integral: \begin{equation} I := \int_{0}^{t}\operatorname{Ai}(-z)^4\, \mathrm{d}z \quad \mathrm{where } \quad t\in\mathbb{R}_{\geq 0}, \: ...
2
votes
2answers
58 views

Evaluating $\int\frac{dx}{(a\sin x+ b\cos x)^2}$, $a\neq 0.$

Could you just show the hint to solve this integral, please?
1
vote
1answer
59 views

Determine whether $-7 + \frac {14} 3 - \frac {28} 9 + \frac {56} {27} + \dots$ converges or not.

Determine whether the series converges or not. $$-7 + \frac {14} 3 - \frac {28} 9 + \frac {56} {27} + \dots$$ This is the Alternating Harmonic Series but I can't see the common ratio here. Any ...
2
votes
3answers
41 views

How do I find a closed form for this recurrence?

$$a_0=0$$ $$a_n=a_{n-1} + 2n^2-n$$ What I have so far, but I don't think it's right: $$x^n = x^{n-1} + 2x^2-x^{n-1}=2x+x^{n-2}-1$$ $$0=-x^{n-1}+x^{n-2}+2x-1$$
4
votes
1answer
137 views

Find $\lim_{a\to \infty}\frac{1}{a}\int_0^{\infty}\frac{x^2+ax+1}{1+x^4}\cdot\arctan(\frac{1}{x})dx$

Find $$ \lim_{a\to \infty} \frac{1}{a} \int_0^{\infty}\frac{x^2+ax+1}{1+x^4} \arctan\left(\frac{1}{x}\right)dx $$ I tried to find $$ \int_0^{\infty} ...
13
votes
2answers
137 views

How to integrate $\int\limits_{0}^{\pi/2}\frac{dx}{\cos^3{x}+\sin^3{x}}$?

I have$$\int\limits_{0}^{\pi/2}\frac{\text{d}x}{\cos^3{x}+\sin^3{x}}$$ Tangent half-angle substitution gives a fourth-degree polynomial in the denominator that is difficult to factor.
1
vote
1answer
47 views

Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous [closed]

Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous. I'd like to see a full solution please!
0
votes
1answer
45 views

Find the closed formula for following problem. [closed]

A student borrows $\$ 3000$ on a bank credit card at a nominal rate of $18$% per year, which is actually charged at a rate of $1.5$% per month. a) If a person does not place any additional charges on ...
3
votes
0answers
50 views
1
vote
0answers
34 views

Closed form expression for $x$

We define the following relation: $$y = \frac{(1+x)^n - 1}{x}$$ Alternatively, $$y = \sum_{k = 1}^n {n \choose k} x^{k-1}$$ Where $n$ is an integer and $x$ and $y$ are positive real numbers. How ...
4
votes
3answers
195 views

Closed form for $\displaystyle \sum_{n=1}^\infty \int_0^1 \frac{x^{n-1}\ln^2(x)\ln(1-x)}{n^2} \,dx$ [closed]

I am trying to get this to equal $\displaystyle-\frac {\pi^a}{b}$ for some positive integers $a$ and $b$ . My efforts so far give: $\displaystyle \sum_{n=1}^\infty \int_0^1 ...
1
vote
0answers
33 views

Classifying functions whose inverse do not have a closed form

My initial question contained about how to classify functions whose integrals and inverses do not have a closed form. But I found this question: How can you prove that a function has no closed form ...
12
votes
3answers
189 views

Evaluate $\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy$

I want compute this integral $$\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy, $$ where $ \left\{ x \right\} $ is the fractional part function. Following PROBLEMA 171, Prueba de a), last ...
0
votes
0answers
21 views

Equation with a depended variable as argument of the tangent function: $x= K+y\tan(\sqrt{y})+y$ [duplicate]

What would you do if you need to find a relation between two variables but the one that you need to find in function of the other is the argument of a tangent (tan(y)) ? For example if I have the ...
-1
votes
1answer
15 views

Solve an equation in which a variable is an argument of a tangent: $x= K+y\tan(\sqrt{y})+y$

What would you do if you need to find a relation between two variables but the one that you need to find in function of the other is the argument of a tangent ($\tan(y)$) ? For example if I have the ...
3
votes
1answer
57 views

Find closed formula for $a_{n+1}=(n+1)a_{n}+n!$

$a_{n+1}=(n+1)a_{n}+n!$ where a0=0 and n>=0. To get the closed form, I'm trying to find an exponential generating function for the above recurrence, but it doesn't seem to be very nice. Am I going ...
3
votes
1answer
54 views

Find closed formula for the recurrence $a_{n}=na_{n-1}+n(n-1)a_{n-2}$

$a_{n}=na_{n-1}+n(n-1)a_{n-2}$ where a0 = 0, a1=1, and n >= 2. I found an exponential generating function for this recurrence, but cant seem to find the closed form because the generating function ...
2
votes
1answer
109 views

Does a closed form solution exist for $x$? [closed]

$$ \sqrt{x} + x^2 = \sqrt{2} $$ If so, how would one find it?
1
vote
0answers
27 views

Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure

I would like to know if there exists a closed form of the Baker Campbell Hausdorff theorem subject to the conditions that $[x,[x,y]] \sim x$ and $[y,[x,y]] \sim y$. The simple cases that I know a ...
8
votes
2answers
126 views

closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

$$I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$$ for $n=1$ I tried to use $\arctan x=u$ and by notice that $$\frac{1+\tan u}{1-\tan u}=\cot\left ( ...
1
vote
1answer
64 views

Closed form of this expression

So at my school today, we celebrated Pi Day, and we had a customary mini-olympiad. In that olympiad there was a really interesting question. There is a sequence: $\lbrace 1\rbrace, \lbrace 4, ...
1
vote
1answer
24 views

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 ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
10
votes
1answer
67 views

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 ...
3
votes
2answers
64 views

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?
2
votes
1answer
92 views

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 $b_n$ ...
1
vote
1answer
31 views

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 $y_n$, ...
5
votes
2answers
330 views

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? ...
4
votes
1answer
48 views

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 ...
4
votes
0answers
42 views

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 ...
2
votes
0answers
50 views

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 ...
1
vote
1answer
27 views

Risch algorithm analogue for differential equations

I know that we can determine whether an integral has closed form, that is, is a composition of elementary functions. That problem is (more or less) solved by Risch algorithm. For differential equation ...
4
votes
2answers
151 views

Closed form 0f $I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \right) } \right) }^{ 3 } }{ 1-x } dx }$

While solving a problem, I got stuck at an integral. The integral is as follows: Find the closed form of: $$I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \right) } ...
8
votes
1answer
192 views

How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$

How to calculate this integral $$\int_0^\pi \ln(1+\sin x)\mathrm dx$$ I didn't find this question in the previous questions. With the help of Wolframalpha I got an answer $-\pi \ln 2+4\mathbf{G}$, ...
0
votes
0answers
108 views

What is close-form solution using ALS algorithm to optimize this objective function

$C \in \mathbb{R}^{m \times n}$, $X \in \mathbb{R}^{m \times n}$, $W \in \mathbb{R}^{m \times k}$, $H \in \mathbb{R}^{n \times k}$, $S \in \mathbb{R}^{m \times m}$, $P \in \mathbb{R}^{n \times n}$ ...
2
votes
2answers
88 views

How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
1
vote
0answers
24 views

Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
8
votes
3answers
122 views

General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ ...