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".
3,693
questions
2
votes
1
answer
84
views
Finding the minimum for this formula
I think if we want to calculate $\min_x \sum_i (b_i - x)^2$, the answer should be the mean of $b_i$, right? Now if we add a weight to each term and make it $\min_x \sum_i w_i(b_i - x)^2$, what's the ...
- 149
6
votes
1
answer
23k
views
How to find the closed form to the fibonacci numbers? [duplicate]
Possible Duplicate:
Prove this formula for the Fibonacci Sequence
How to find the closed form to the fibonacci numbers?
I have seen is possible calculate the fibonacci numbers without recursion, ...
- 91
2
votes
3
answers
203
views
Find a closed term for $f(n) = n + 2 f(n-1)$, $f(1)=1$
I cannot help myself, but I don't get the closed term for: $f(n) = n + 2 f(n-1)$, where f(1) = 1. I tried to find the pattern when looking at some iterations, and I think I see the pattern very ...
- 31
4
votes
4
answers
843
views
How to get closed form from generating function?
I have this generating function:
$$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{
\frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$
and I know that $\frac {1}{\sqrt {1-4\,z}}$ is ...
- 101
17
votes
2
answers
727
views
Who has the upper hand in a generalized game of Risk?
So, I played a game of Risk the other day for the first time since I was very little. I was frustrated to discover that I couldn't compute (at least not in my head) whether the attacker or the ...
- 22.4k
4
votes
2
answers
150
views
Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?
Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$:
$$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$
$a$ and $b$ are positive constants.
...
- 41
17
votes
3
answers
3k
views
closed form of $\sum \frac{1}{z^3 - n^3}$
I am currently trying to find a closed form expression for $\displaystyle f(z) = \sum_{n \in \mathbb{Z}} \frac{1}{z^3 - n^3}$, $z \in \mathbb{C}$. After a set of twists and turns, I have hit a wall.
...
- 92.6k
12
votes
2
answers
315
views
The value of the series $\frac11+\frac1{2+3}+\frac1{4+5+6}+\cdots$ and $\frac{1}{1}+\frac{1}{2\cdot3}+\frac{1}{4\cdot5\cdot6}+\cdots$
What is the value of the sum of the series
$$\frac{1}{1}+\frac{1}{2+3}+\frac{1}{4+5+6}+\cdots\;?$$
And this:
$$\frac{1}{1}+\frac{1}{2\cdot3}+\frac{1}{4\cdot5\cdot6}+\cdots\;?$$
- 8,759
1
vote
2
answers
210
views
Are there sequences in which you can prove there are no closed form?
Any finite sequences can be expressed as a polynomial, but there are many infinite sequences for which we have found no closed form. Is it possible that no closed form exists? Are there sequences in ...
- 587
0
votes
1
answer
891
views
Rearranging a general closed form linear recurrence sequence
I have the following general closed form linear recurrence equation:
$$x_n=r^{n-1}a+\left(\frac{r^{n-1}-1}{r-1}\right)d, \qquad (n=1,2,3,...)$$
and the next stage in the text shows the equation ...
- 521
2
votes
2
answers
151
views
Representation of this function using a single formula without conditions
Is it possible to represent the following function with a single formula, without using conditions? If not, how to prove it?
$F(x) = \begin{cases}u(x), & x \le 0, \ v(x) & x > 0 \end{cases}...
- 177
6
votes
4
answers
234
views
closed form for $d(4)=2$, $d(n+1)=d(n)+n-1$?
I am helping a friend in his last year of high school with his math class. They are studying recurrences and proof by inference. One of the exercises was simply "How many diagonals does a regular $n$-...
- 349
63
votes
6
answers
20k
views
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
- 26.4k
1
vote
1
answer
362
views
Closed-form solution for this system of ODEs
I am trying to solve the following system (derived from a Michaelis-Menten kinetics model for an enzymatic chemical reaction):
$$\dot{y}_a = r_p x_a - \lambda_p y_a$$
$$\dot{x}_b = \frac{\alpha_0 + \...
- 255
2
votes
2
answers
705
views
Closed-form Expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$? (problem with Mathematica)
I need to calculate a closed-form expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$. This isn't particularly difficult, and I do it by hand pretty much routinely.
However I found out ...
- 364
2
votes
2
answers
905
views
A double integral (differentiation under the integral sign)
While working on a physics problem, I got the following double integral that depends on the parameter $a$:
$$I(a)=\int_{0}^{L}\int_{0}^{L}\sqrt{a}e^{-a(x-y+b)^2}dxdy$$
where $L$ and $b$ are constants.
...
- 6,908
61
votes
13
answers
161k
views
What is the solution of $\cos(x)=x$?
There is an unique solution with $x$ being approximately $0.739085$. But is there also a closed-form solution?
- 1,055
4
votes
1
answer
141
views
Two questions related two series and convergence
1)
I applied Raabe's test on both and guessed the answer as (B), but not convinced enough. Is there a better approach?
2) Is there any closed form of following the series. It is however, known that ...
- 613
2
votes
2
answers
986
views
Generating Function explicit formula
Say i got
$\displaystyle{\frac{(1-2x)}{(1+3x)^3}}$
I used $\displaystyle{\frac{1}{(1+3x)}}$ $=\sum_{n=0}^\infty(-3)^n x^n$ and differentiated twice
I got $\displaystyle{\frac{(1-2x)}{(1+3x)^3}}$ = ...
- 33
1
vote
2
answers
459
views
Looking for a closed form to determine whether a symbol is part of the ith combination nCr
Hi I'm new to this, feel free to correct or edit anything if I haven't done something properly.
This is a programming problem I'm having and finding a closed form instead of looping would help a lot.
...
- 11
5
votes
1
answer
409
views
Power Mean Random Distribution
I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$.
I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor nx\rfloor}(-1)^k\binom{n}{k}(nx-k)^{n-1}...
- 4,814
31
votes
6
answers
3k
views
Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
Today, we had a math class, where we had to show, that $a_{100} > 14$ for
$$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$
Apart from this task, I asked myself: Is there a closed form for this ...
- 9,337
3
votes
1
answer
278
views
Closed forms of sums $f(a)+f(a+d)+\cdots+f(a+nd)$ with $f$ sine, cosine or tangent
is/are there a closed form for
$\sin{(a)}+\sin{(a+d)}+\cdots+\sin{(a+n\,d)}$
$\cos{(a)}+\cos{(a+d)}+\cdots+\cos{(a+n\,d)}$
$\tan{(a)}+\tan{(a+d)}+\cdots+\tan{(a+n\,d)}$
$\sin{(a)}+\sin{(a^2)}+\...
- 691
3
votes
2
answers
630
views
expression for the sum involving digamma function
I got this answer from WolframAlpha. Does anyone know how even to approach it to obtain the solution using digamma function. Please don't solve it, just show me in the right direction!
$$
\sum_{k=1}^...
- 1,727
5
votes
6
answers
455
views
Find the sum $\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$
How to find this sum?
$$\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$$
- 173
2
votes
1
answer
382
views
Closed form for $\sum_{k=0}^{n} \cos( t \sqrt{k} )$?
I would like to know if there a closed form solution for the sum:
$$ S_n(t) = \sum_{k=0}^{n} \cos( t \sqrt{k} ) $$
There is obviously an easy answer when the sum is replaced by an integral so this ...
- 1,020
2
votes
2
answers
573
views
How to find a closed form for a sum involving $\max(x,y)$
I have this sum:
$$\sum_{0\le y<k}\sum_{0\le x<k-y}k - \max(x,y)\ ,\qquad k\in\mathbb{N}$$
Is there a closed form for it? This is no homework, im just a highschool student whose math is too ...
- 9,337
3
votes
0
answers
309
views
How do they know an integral has no closed form solution? [duplicate]
Possible Duplicate:
How can you prove that a function has no closed form integral?
When they say that, e.g., Li(x) has no closed form (for some agreed upon definition of "closed form"), do they ...
- 645
47
votes
6
answers
24k
views
What does closed form solution usually mean?
This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form ...
- 38.8k
2
votes
2
answers
359
views
Identify this power series / solve this trig equation
I was asked to find a solution to
$$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$
where $n$ is a fixed integer greater than 1.
Numerically, there's a solution just above 1/n so I decided to find this ...
- 32.3k