All Questions
20
questions
6
votes
1
answer
143
views
Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.5k
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ 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} $$
I want to find ...
- 756
1
vote
2
answers
83
views
Evaluating $\lim_{k\to+\infty}\frac12\sin(\sqrt{k+1})-2 \left(\sin(\sqrt{k+1})-\sqrt{k+1}\cos(\sqrt{k+1})\right)+\sum_{n=0}^k\sin(\sqrt n)$
I need help at evaluating this to some closed form formula:
$$\lim_{k\to+\infty}\frac{\sin\left(\sqrt{k+1}\right)}{2}-2 \left(\sin\left(\sqrt{k+1}\right)-\sqrt{k+1}\cos\left(\sqrt{k+1}\right) \right)+\...
- 21
5
votes
1
answer
115
views
Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?
On the graph of $y=\tan{x}$, $0<x<\pi/2$, draw $2n$ zigzag line segments that, with the x-axis, form equal-width isosceles triangles whose top vertices lie on the curve. Here is an example with $...
- 25.8k
2
votes
1
answer
91
views
Closed form of the series: $\sum _{n=0}^{\infty }\left(\frac{1}{2^n\left(1+\sqrt[2^n]{x}\right)}\right)$
Came across this in a calc textbook from the 1800s and I can't figure out a way to solve it. Trying to write it in product form by taking the integral didn't work. I also tried adding consecutive ...
- 113
5
votes
1
answer
254
views
Evaluate $\lim\limits_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\sin^n(\frac{1}{n})}{\frac{1}{1!}+\frac{1}{2!}+\ldots+\frac{1}{n!}}$
This was a recent problem on the Awesome Math Problem Column. The solution is given as follows:
We shall use Stolz-Cesaro Lemma. We have:
$$\lim_{n\to\infty} \frac{\sin(1)+\sin^2(\frac{1}{2})+\ldots+\...
- 519
0
votes
1
answer
96
views
Find a recursive formula for a closed formula recursively at infinity
I have a recursive sequence defined as such:
$$
\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
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
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.
$$\begin{align}\...
- 30.8k
2
votes
1
answer
57
views
Closed form and limit of the sequence $a_{n+1}=\frac{-5a_n}{2n+1}$
I have no idea about how to deal with point B. Can anyone help me? Also, an elegant way to solve point A would be great but it's not that important. Thanks in advance for the help!
A) Suppose $a\in\...
- 31
4
votes
0
answers
162
views
Determine sum of the series $\sum_{k=1}^\infty\frac{1}{n(n+1)(n+2)}$ [duplicate]
I have the following problem,
$$\sum_{k=1}^\infty\frac{1}{n(n+1)(n+2)}$$
And I try to work as follows:
Hint: Partial Fraction decomposition:
$\begin{aligned}
\frac{1}{n(n+1)(n+2)} &= \frac{...
- 3,919
3
votes
2
answers
90
views
Evaluating the limit of the sequence: $\frac{ 1^a + 2^a +..... n^a}{(n+1)^{a-1}[n^2a + n(n+1)/2]}$
My friend gave me this question to solve a few days ago and after I got no way to solve this, I thought I should seek some help.
I had to evaluate the limit of the following when $n$ tends to ...
1
vote
1
answer
59
views
Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $
As the question says,
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$
where a is a constant, $a>0$.
