All Questions
Tagged with closed-form limits
71
questions
0
votes
1
answer
96
views
Find a recursive formula for a closed formula recursively at infinity
I have a recursive sequence defined as such:
$$
\left(u_k \right) = \begin{cases}
u_0 = 1 \\
u_k = u_{k-1} + u_{k-1} \cdot \frac{1}{n}
\end{cases}\quad \text{with}\...
- 936
1
vote
0
answers
77
views
About $\sqrt[k]{l + \sqrt[k]{l + \sqrt[k]{l + ...}}} $ asymptotics
Consider simple nested radicals
More precisely Let
$$ 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
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
2
answers
167
views
$ a_n = \frac{a_{n-1}(a_{n-1} + 1)}{a_{n-2}}.$ and $ T = 3.73205080..$?
Consider the following sequence :
Let $a_1 = a_2 = 1.$
For integer $ n > 2 : $
$$a_n = \frac{a_{n-1}(a_{n-1} + 1)}{a_{n-2}}.$$
$$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$
$$T = ??$$
...
- 16.4k
2
votes
1
answer
938
views
A limit of combination
I want to find the closed form of the limit,
\begin{align*}
I(k,r):=\lim_{x\rightarrow 0}\left\{\sum\limits_{j=1}^{r+2-k} (-1)^{r+3-j-k} \binom{r-j}{k-2}\frac{1}{x^j}+\frac{1}{(1+x)^{k-1}x^{r-k+2}}\...
- 537
2
votes
3
answers
407
views
Exact value of the limit of the Mandelbrot iteration for c=1/4?
I know and understand the proofs, that c $\to c^2+c \to(c^2+c)^2+c \to ...$ converges, if $0<=c<=\frac{1}{4}$. But I wonder: is the exact value (a closed expression) for the limit known for the ...
- 75
1
vote
0
answers
79
views
Find the limit: $\lim_{n\to\infty} \left (1+ \frac {1}{2^2}+ \frac {1}{3^3}+...+\frac {1}{n^n} \right)$ [duplicate]
Does the limit:
$$\lim_{n\to\infty} \left (1+ \frac {1}{2^2}+ \frac {1}{3^3}+...+\frac {1}{n^n} \right)$$
admit a closed form?
I know only Riemann-Zeta function.I've just discovered this sum.
- 477
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
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
39
votes
3
answers
2k
views
What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?
Let's look at the continued radical
$ R = \sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $
whose signs are defined as $ (+, -, +, -, -, + ,-, -, -,...)$, similar to the sequence $...
user346361
8
votes
2
answers
743
views
Why is $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots = \frac{\pi}{4}$? [duplicate]
I was watching a numberphile video, and it stated that the limit of $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$ was $\frac \pi 4.$
I was just wondering if anybody could prove this using some ...
- 3,424
3
votes
1
answer
60
views
strategies to find explicit formulae for series
I have been manipulating a certain series for several hours without finding any pattern. Hence I am wondering what some of the better strategies are to find patterns and thus an explicit formula for a ...
- 1,363
4
votes
2
answers
128
views
Let $\sum_{n=0}^\infty \frac{(-1)^{n+1}}{3 n+6 (-1)^n}$, does it converge or does it diverge?
Let $\displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}}{3 n+6 (-1)^n}$, does it converge or does it diverge?
I'm not completely sure that my calculation is correct, check it please.
$$\begin{align}\...
- 30.8k
2
votes
1
answer
57
views
Closed form and limit of the sequence $a_{n+1}=\frac{-5a_n}{2n+1}$
I have no idea about how to deal with point B. Can anyone help me? Also, an elegant way to solve point A would be great but it's not that important. Thanks in advance for the help!
A) Suppose $a\in\...
- 31
6
votes
2
answers
144
views
2 conjectured recursion limits for $e$ and $\pi$. [duplicate]
Consider the following recursions
$$ x_{n+2} = x_{n+1} + \frac{x_n}{n} $$
$$y_{n+2} = \frac{ y_{n+1}}{n} + y_n $$
I have been toying around with different starting values ( complex Numbers ) , ...
- 16.4k