All Questions
Tagged with closed-form limits
13
questions with no upvoted or accepted answers
34
votes
0
answers
597
views
An iterative logarithmic transformation of a power series
Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion:
$$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$
Then, at each step ...
- 47.3k
5
votes
0
answers
131
views
Infinite product of areas in a square, inscribed quarter-circle and line segments.
The diagram shows a square of area $An$ and an enclosed quarter-circle.
Line segments are drawn from the bottom-left vertex to points that are equally spaced along the quarter-circle.
The regions ...
- 25.8k
4
votes
0
answers
72
views
How do I find the finite limits of this infinite product?
What is... $$\lim_{\omega \to \infty}
\left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$
I'd like closed form solutions, and in this case that means any ...
- 8,361
3
votes
0
answers
121
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Special property of circle with radius 0.975399...
$4n$ points are uniformly distributed on a circle. Parabolas are drawn in the manner shown below with example $n=4$.
The parabolas' vertices are at the center of the circle. The parabolas have a ...
- 25.8k
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find ...
- 756
2
votes
0
answers
264
views
Closed form for $\sqrt{1+\sqrt{x+\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+...}}}}}$
Inspired by this question that I recently saw, I was wondering if there is a closed form for $$y = \sqrt{1+\sqrt{x+\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+...}}}}}$$
as a function of $x$.
Usually, in problems ...
- 8,917
2
votes
4
answers
155
views
Find closed formula and limit for $a_1 =1$, $2a_{n+1}a_n = 4a_n + 3a_{n+1}$
Tui a sequence $(a_n)$ defined for all natural numbers given by
$$a_1 =1, 2a_{n+1}a_n = 4a_n + 3a_{n+1}, \forall n \geq 1$$
Find the closed formula for the sequence and hence find the limit.
Here, ...
- 157
1
vote
0
answers
72
views
Fibonacci like sequence $f(n) = f(n-1) + f(n-2) + f(n/2)$ and closed form limits?
Consider
$$f(1) = g(1) = 1$$
$$f(2) = A,g(2) = B$$
$$f(3) = 1 + A,g(3) = 1+B$$
And for $n>3$ :
$$f(n) = f(n-1) + f(n-2) + f(n/2)$$
$$g(n) = g(n-1) + g(n-2)$$
where we take the integer part of the ...
- 16.4k
1
vote
0
answers
100
views
Reduction of $_3\text F_2(a,a,1-b;a+1,a+1;x)$ with the hypergeometric function
A derivative of the incomplete beta function $\text B_x(a,b)$ uses hypergeometric $_3\text F_2$
$$\frac{d\text B_x(a,b)}{da}=\ln(x)\text B_x(a,b)-\frac{x^a}{a^2}\,_3\text F_2(a,a,1-b;a+1,a+1;x)$$
Now ...
- 12.5k
1
vote
0
answers
77
views
About $\sqrt[k]{l + \sqrt[k]{l + \sqrt[k]{l + ...}}} $ asymptotics
Consider simple nested radicals
More precisely Let
$$ K > 1 , 1 \leq l $$
$$X(j,K) = X_\infty(j,K)$$
$$X_0(j,K) = a(j,K)$$
$$X_n(j,K) = \sqrt[k]{j + X_{n-1}(j,K)}$$
$$Y(j,K) = \frac{j + X_{\...
- 16.4k
1
vote
0
answers
205
views
Question on series being expressed in closed form
Given an integer $k$ and $0\leq \alpha \leq 1$, let $f_1(\alpha)=1/k$ and $f_{i+1}(\alpha)=\frac{(k-1)f_i(\alpha) + (f_i(\alpha)^{1/\alpha} + 1)^\alpha}{k}$.
Consider the function $g(\alpha) = \lim_{...
- 11
0
votes
0
answers
50
views
Asymptotics for this limit iteration with $f(x)= 2x + x^5 ,g(x) = x + x^3$
Consider $x>0$
Let
$$f(x)= 2x + x^5$$
$$g(x) = x + x^3$$
$$f(r(x))=r(f(x))=id(x)$$
$$g(u(x)) = u(g(x))=id(x)$$
Where $id(x)$ is the identity function mapping all values to itself.
Let $*^{[y]}$ ...
- 16.4k
0
votes
0
answers
97
views
Expressing the Golomb-Dickman constant in closed-form
Is there a way to express the Golomb-Dickman constant ($\lambda$) (A084945) in a closed-form expression? Here's the Wikipedia article for the Golomb-Dickman constant, but it's not as useful in my ...
- 1,271