Tagged Questions
10
votes
6answers
110 views
A limit on binomial coefficients
Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.
What I can do is just use Stolz formula. But I could not proceed.
7
votes
3answers
248 views
How to calculate $\sum \limits_{x=0}^{n} \frac{n!}{(n-x)!\,n^x}\left(1-\frac{x(x-1)}{n(n-1)}\right)$
What are the asymptotics of the following sum as $n$ goes to infinity?
$$
S =\sum\limits_{x=0}^{n} \frac{n!}{(n-x)!\,n^x}\left(1-\frac{x(x-1)}{n(n-1)}\right)
$$
The sum comes from Probability of ...
7
votes
2answers
139 views
limit of the sum $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $
Prove that : $\displaystyle \lim_{n\to \infty} \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{2n}=\ln 2$
the only thing I could think of is that it can be written like this :
$$ ...
4
votes
1answer
157 views
Limit to infinity of a summation
I'm just beginning my journey with calculus, and this problem is giving me hard time.
$$\lim_{n \to \infty} \sum_{k=1}^n \frac{k}{{n^2+k}}$$
I calculated first three sums: $$\frac{1}{2}, ...
4
votes
1answer
51 views
Limit of difference of integral and sum
$f:[0,1]\rightarrow\mathbb R$ and $f\in C^1$, then the limit $\lim_{n\rightarrow\infty} n(\int_{0}^{1}f(x)dx-\frac{1}{n}\sum_{k=1}^{n}f(\frac{k-1}{n}))$ exists.
I guess the kernel lies in the sum ...
3
votes
2answers
165 views
A limit about euler's constant
Show that :
$$\lim_{m\to \infty}\left[ -\frac{1}{2m}+\ln \left( \frac{\text{e}}{m} \right)+\sum\limits_{n=2}^m \left( \frac{1}{n}-\frac{\zeta \left( 1-n \right)}{m^n} \right) \right]=\gamma $$
How to ...
3
votes
6answers
123 views
Convergence of the series
Im trying to resolve the next exercise:
$$\sum_{n=1}^\infty\ e^{an}n^2 \text{ , }a\in R $$
I dont know in which ranges I should separe the a value for resolving the limit and finding out the ...
3
votes
3answers
46 views
Help finding the limit of a sum
Hi I'm trying to find the following limit:
$$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{ n } (1 - e^{\frac{-jt}{n}} )$$
expressed as a funciton of t. You may even be able to get it from ...
3
votes
0answers
79 views
Limit involving sums of the Von-Mangoldt function
Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem?
$$ \lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) ...
2
votes
2answers
48 views
Show convergence and calculate the limit
I guess it has something to do with riemann sums but this is new for me.
$\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$
How do i start?
2
votes
3answers
44 views
Sum of $\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n} $
Calculate the sum of the next series and for which values of $x$ it converges:
$$\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n}$$
I used D'Alembert and found that the limit is less than 1, so: $-5 < x ...
2
votes
1answer
70 views
Having difficulty with Summation
How would I compute:
$$\sum_{n=2}^\infty \frac{1}{n^2 - n} \cdot n$$
Hints or step by step process would be the most helpful.
2
votes
1answer
131 views
How to prove that $\frac1{\sqrt{n}}\sum\limits_{k=2}^{n+1} \prod\limits_{\ell=1}^{k-2}\frac{n-\ell}{n}$ converges to $\sqrt{\pi/2}$?
Consider $$
X_k =\prod_{\ell=1}^{k-2}\frac{n-\ell}{n}
\ \textrm{ for every } 2\leqslant k\leqslant n+1.
$$
How can you prove the following?
$$
\lim_{n\rightarrow \infty} ...
2
votes
2answers
151 views
Convergence of $\sum_{i=2}^{\infty}\frac{1}{i \cdot f(i)}$
[Edited to fix typo]
Is there a precise formulation for when the sum
$$
\sum_{i=2}^{\infty}\frac{1}{i \cdot f(i)}
$$ converges, in terms of the function $f$? Assume that $f$ is smooth and ...
1
vote
2answers
76 views
The mathematical and technical approach to a limit of a sum of a sequence
In Calculus, what is the most preferred mathematical and technical way to approach a limit of a sum of a sequence?
Take for example:
$$ \lim_{n \to \infty} {\frac{1}{n} \sum_{k=1}^{n} {\ln(1 + ...
