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

learn more… | top users | synonyms

0
votes
1answer
56 views

Find $\lim\limits_{x\to0}\left(\frac{1}{\tan(x)}-\frac{1}{{e^x-1}}\right)$ [closed]

Please, help to find this limit. $$\lim\limits_{x\to0}\left(\frac{1}{\tan(x)}-\frac{1}{{e^x-1}}\right)$$
1
vote
1answer
46 views

Solving a particular nonlinear recurrence relation

I am trying to solve the recurrence relation $a_{n}=\alpha a_{n-1}^2+\beta a_{n-1}$ where $\alpha$ and $\beta$ are constants. I have been trying to find specific techniques for solving this equation ...
6
votes
4answers
225 views

A limit related to super-root (tetration inverse).

Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ ...
3
votes
1answer
122 views

Is there a proof that the Harmonic numbers are not an elementary function? [duplicate]

The Harmonic numbers $H_x = \sum_{n=1}^x 1/n$ are the sum of the reciprocals of the natural numbers up to a given number. The first few are $0, 1, 3/2, 11/6, \ldots$. $H_x$ can be defined for ...
2
votes
2answers
84 views

Proving the closed form of $\sin48^\circ$

According to WA$$\sin48^\circ=\frac{1}{4}\sqrt{7-\sqrt5+\sqrt{6(5-\sqrt5)}}$$ What would I need to do in order to manually prove that this is true? I suspect the use of limits, but I don't know where ...
1
vote
0answers
158 views

No closed form for $\sum_{n\in P} \frac{1}{n^2}$

I think that I can say with a fair amount of assurance that $$\sum_{n\in \mathcal P} \frac{1}{n^2}$$ has no closed form (assuming that $\mathcal P$ represents the full set of primes) I currently know ...
7
votes
1answer
218 views

How to evaluate $\int_0^{\pi /2}\frac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du$?

I want to find the value of $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du.$$ Let $v=\frac{\pi}{2}-u$, then $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos ...
0
votes
3answers
38 views

Closed form solution for the recurrence

I am given the following recurrence and need to find a closed form solution for the recurrence. I have no idea on how to get started though and i need some help on leading me to solve this. $A_0=20, ...
0
votes
1answer
72 views

Can I have some assistance with this integral calculation?

This integral has bothered me for the longest time: $$J=\int_{-1}^0 \sqrt[x]{2+\Gamma(x+1)}\space\text{dx}$$ This guy is extremely minuscule in relation to most other integrals but was amazingly ...
21
votes
0answers
360 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
1
vote
2answers
38 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
1
vote
0answers
27 views

What formulas are available to find the nth digit of a number?

Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have, $$(1) \quad \operatorname{d_n}(x)=\lfloor ...
1
vote
1answer
41 views

Closed form for the first local min $\gt 0$ of $x!$ (in reference to x-value of min)

The first local min of $x!$ is the point $(0.461632...,0.885603...)$ Is there a close form of $0.461632...$, the $x$-value of the above point? If you can tell me the closed form, could you help me ...
6
votes
1answer
83 views

Game in a circle

$N$ players play a game. They stand in a way such that they form a regular $N$-gon. Players are numbered from $1$ to $N$. The players throw boomerangs in clockwise order, in turns. At first player $1$ ...
1
vote
1answer
62 views

Solving a tricky recurrence relation

Given the following recurrence relation: $T_2=1$ $T_4=4$ $T_{2n}= \begin{cases} T_{2n-2}+3\bmod2n & 2T_{2n-2}\geq2n-2\\ T_{2n-2}+2\bmod2n & 2T_{2n-2} < 2n-2\\ \end{cases} ...
-1
votes
1answer
51 views

Evaluate the sum $\sum_{k=1}^nk^2\, 2^{-k}$

By sum properties, prove that: $$\sum_{k=1}^nk^2\, 2^{-k}=2^{-n}(-6+3\cdot 2^{1+n}-4n-n^2)$$ Progress so far: $$\sum_{k=1}^nk^2\,2^{-k} = 1\cdot (1/2) + 4\cdot (1/4) + 9\cdot (1/8) + 16\cdot ...
6
votes
1answer
70 views

Summation of a series involving powers of Fibonacci numbers.

I'm interested in this series: $$\mathcal S_p=\sum_{n=1}^\infty\frac{\left(F_n\right)^p}{2^{np}},\quad p\in\mathbb N,\tag1$$ where $F_n$ are the Fibonacci numbers, defined by the recurrence ...
4
votes
1answer
128 views

Prove that $\int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx = 1$

While I've been thinking about this question, I've found that for all $n \geq 1$ integer values, we have $$ \mathcal{I}_n = \int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx ...
2
votes
0answers
33 views

Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$ \log f(x) + g(x) = c $$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
2
votes
1answer
57 views

Reference to proof of closed form for of $\zeta_{\text{Ai}}(2)$

I was stunned to find out we can evaluate the Airy Zeta Function at $s = 2$ exactly: $$ \zeta_{\text{Ai}}(2) \equiv \sum_{k=1}^\infty a_k^{-2}=\frac{3^{5/3}}{4\pi^2}\Gamma^4\left(\frac23\right) $$ ...
2
votes
2answers
56 views

Closed form of $\exp(\sum_{n = 1}^{\infty} \frac{t^n}{n})$

I want to show that $$\exp(\sum_{n = 1}^{\infty} \frac{t^n}{n}) = \frac{1}{1 - t},$$ for $t \in (-1, 1)$. Using derivatives and fundamental theorem of calculus, I have a basic idea how to proceed. ...
3
votes
0answers
49 views

Any closed form for this expression?$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$

I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank! $$ ...
3
votes
1answer
57 views

How to solve a functional differential equation?

$$(1) \quad \cfrac{d}{dx} (f(x^n))=\cfrac{-f(x^n)^2}{f(n \cdot x^{n-1})}$$ How do I solve this functional differential equation? I need a closed form solution, so approximations won't cut it, I'll ...
3
votes
2answers
71 views

Improper integral of $\frac{\sin x}{x}e^{-ax}$

For $a>0$ define $$I(a)=\int_0^\infty \frac{\sin x}{x}e^{-ax} \, dx,$$ I can show it is continuous at $0$, but by differentiating in $a$, I can't see why $$I(a)=\frac{\pi}{2}-\arctan(a).$$ Thanks ...
0
votes
1answer
74 views

Finding a closed form of recursive formula $T(n)=4T(n-1) - 4T(n-2)$

Find the closed form for the following: $$T(n) = \begin{cases} 1\quad &\text{ if } n = 0 \\ 4\quad &\text{ if } n = 1 \\ 4T(n-1) - 4T(n-2) & \text{ if } n > 1 \end{cases}$$ ...
0
votes
0answers
31 views

Looking for rounded corner plane curve with certain properties (SIDESTEPPED)

For a project involving simulating traffic lights, I am currently looking for a formula to get a rounded 90-degree corner (to describe the path of a turning car) with certain properties: Defined in ...
3
votes
1answer
153 views
2
votes
1answer
66 views

Closed form for ${\large\int}_0^1x\,\operatorname{li}\!\left(\frac1x\right)\ln^{1/4}\!\left(\frac1x\right)dx$

Let $\operatorname{li}(x)$ denote the logarithmic integral: $$\operatorname{li}(x)=\int_0^x\frac{dt}{\ln t}.$$ How can we prove the following conjectured closed form? ...
8
votes
3answers
125 views

Is there a closed form for the nontrivial solutions of $x^y = y^x$?

It can be shown $\forall x \in \Bbb{R}^+ \exists \{y_1, y_2\} : x^y = y^x.$, and the first of these two numbers is trivial, $y_1 = x$. The second is nontrivial, and I cannot find a closed form for all ...
3
votes
2answers
78 views

Finding $S=\sum_2^{\infty}\frac{\ln(n+1)}{(n^2-1)}$

Is there any chance to express the series $$S=\sum_2^{\infty}\frac{\ln(n+1)}{(n^2-1)}$$ in terms of a known function? My idea is to start from the generalized-Euler-constant function ...
5
votes
1answer
76 views

Closed form for finite sum of ${\rm csch}^2$

In a recent problem I was attempting to solve, I hit a road block when I reduced the problem to that of finding a closed form for the following sum $$ \mathcal{S}_n(x)\equiv\sum_{k=1}^n{\rm ...
4
votes
4answers
121 views

Compute a double integral: $\int_0^1\int_0^1 \frac{(xy)^s}{\sqrt{-\log(xy)}}dxdy$

Is it possible to get a closed form of the following integral? $$I=\int_0^1\int_0^1 \frac{(xy)^s}{\sqrt{-\log(xy)}} \, dx \, dy\quad \quad \quad (s>0).$$ My attempt: I’ve tried a change of ...
1
vote
1answer
51 views

Can this integral be computed in closed-form?

I am interested in computing a normalizing constant (of a Gaussian density in dimension $3$). Such normalizing constants often do not have a closed form. In dimension $2$, this normalizing constant ...
5
votes
1answer
160 views

Strategies for evaluating sums $\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}$

I'm looking for strategies for evaluating the following sums for given $z$ and $m$: $$ \mathcal{S}_m(z):=\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}, $$ where $H_n^{(m)}$ is the generalized harmonic ...
5
votes
1answer
110 views

Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

Does anyone know if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ For low values of $a$, Wolfram Alpha gives a closed form in terms ...
-1
votes
1answer
43 views

Is it possible to find a closed form for this recursive sequence?

Is it possible to find a closed form for the sequence defined by $$\begin{align*} a_0&=3\\ na_n&=(n-1)a_{n-1}+1\quad\text{for }n\ge 1\;. \end{align*}$$
0
votes
1answer
56 views

Closed-form solution for $x^a = (1-x)^{1-a}\cdot b$ with $0 < a < 1$

How can I tell if $x^a = (1-x)^{1-a} \cdot b$ has a closed form solution for $x$, assuming $0<a<1$. It does in the case of $a=\frac{1}{2}$, but is this the only case? Computer algebra systems ...
6
votes
1answer
68 views

Closed form for $\int_0^R \frac{dx}{\sqrt{\ln(1+x)}}$, R>0

I stumbled on an interesting integral doing some physics exercise which did not require its closed form (if it has any). It has, however, sparked my interest and I tried my best to find it, but I ...
3
votes
1answer
52 views

Is there any closed form of the following nested series?

I am wondering if there is any closed form of the following summation? $\sum \limits_{i=0}^{\infty} (q^i \sum \limits_{j=0}^i \dfrac{a^j}{j!})$ where |q|<1. I know that $\sum \limits_{i=0}^{\infty} ...
9
votes
1answer
136 views

Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$

$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ? Inspired by this frequently asked question, ...
0
votes
1answer
40 views

Convergence and sum of an infinite series: $\sum_{i=1}^{\infty}\frac{6}{24 i-4 i^2-35}$

Determine whether the following series is convergent or divergent. If convergent, find the sum. $$\sum_{i=1}^{\infty}\frac{6}{24 i-4 i^2-35}$$ Since the limit of the series is zero, I know that it is ...
1
vote
1answer
113 views

Evaluate the sum $\sum_1^n{2k+1\over k(k+1)}$ in closed form

Evaluate the sum $$\sum_{k=1}^n{2k+1\over k(k+1)}$$ in closed form. You are allowed to have the harmonic number $$H_n = \sum_1^nk^{-1}$$ in your closed–form formula. I am having trouble with this ...
7
votes
1answer
94 views

The value of $\sum_{n=0}^{\infty} \, \bigl(\prod_{i=0}^{n-1} q^n-q^i\bigr)^{-1}$

Let $q > 1$. What can we say about the value of $$\sum_{n=0}^{\infty} \, \bigl(\prod\limits_{i=0}^{n-1} q^n-q^i\bigr)^{-1} ~~?$$ The series clearly converges. Is there a closed form or something ...
0
votes
1answer
161 views

Climbing a n-stair staircase, taking 2 or 3 stairs each step…

Suppose a person has a n-stair staircase to climb, and they can go up exactly 2 or 3 stairs each time they take a step. Generate some initial data. Find and explain the recurrence relation to ...
13
votes
3answers
756 views

Why is there no general form for the harmonic numbers?

The Harmonic numbers $H_n$ are given by the sum of the reciprocals of the natural numbers up to a given $n$, ie: $H_1 = 1$ $H_2 = 1 + 1/2 = 3/2$ $H_3 = 1 + 1/2 + 1/3 = 11/6$ $H_n$ for noninteger ...
0
votes
1answer
64 views

Need help finding the closed form of a sequence based upon the fibonacci sequence.

I have been given an assignment question that asks for a simple closed form of the following sequence: $$G_n=\left|\begin{array}{cc} F_n & F_{n+1}\\ F_{n+1} & F_{n+2} \end{array}\right|$$ I ...
1
vote
0answers
31 views

From definite integral to closed-form expression (if the “conversion” exists)

How do I "convert" any definite integral to a closed-form expression (if it exists)? Keep in mind I am pretty new to this area. The main reason I've "stumbled across" this area is because I am looking ...
5
votes
3answers
175 views

Closed form of: $\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $

$\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $ Does a closed form of the above integral exists? $n$ is a positive integer
8
votes
4answers
172 views

Find the solution of $\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$

Is anyone able to help me with the following equation concerned the floor function $\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$ I don't know how to deal with the floor terms properly.