Tagged Questions
10
votes
6answers
113 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.
8
votes
1answer
162 views
Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$
Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$
I don't understand where to start. Please help.
9
votes
3answers
198 views
Problem of limit with binomial coefficients
I thought that the following would made a nice exercise, but I am not sure how to evaluate its difficulty since I often miss elementary solutions. How about you try answering it? It would be great to ...
5
votes
2answers
67 views
Prove that $\lim_{n \to \infty} \binom{n}{k}a^n = 0$
I'm working with this problem but I have no idea how to solve it. Here $k$ is fixed and $0<a<1$.
I was trying to use that $\lim_{n \to \infty} a^n =0$ and that $\binom{n}{k}\leq\frac{n^k}{k!}$ ...
0
votes
1answer
110 views
Limit with binomial coefficients
I am trying to compute the following limit (k is a fixed constant):
$$ \lim_{n\to\infty} \frac{ {n/2 - 1\choose(k-1)/2} {n/2 \choose (k-1)/2} }{n-1 \choose k-1} $$
I expanded the binomial coefficient ...
6
votes
3answers
159 views
What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$?
What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$ in terms of $d$? Does the limit exist? Is there a simple upper bound interms of $d$?
1
vote
2answers
195 views
Limit of binomial coefficient
I would like to find the limit
$$
\lim_{n \to \infty} \binom{s}{n+1} = \lim_{n \to \infty} \frac{s (s-1) \cdots (s-n)}{(n+1)!} ,
$$
where $s \in \mathbb C$.
Actually, it would be enough to show that ...
10
votes
3answers
250 views
Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$
I try to compute the following limit:
$$\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$$
I'm interested in finding some reasonable ways of solving the limit. I don't find any ...
3
votes
1answer
245 views
Binomial fraction sum to infinity
Compute the limit:
$$\lim_{n\to\infty} \sum_{k=0}^n \frac {\dbinom{n}{k}}{\dbinom{2n-1}{k}}$$
Here i tried to give some k values to the sum hoping to see a possible pattern,
but i didn't figure out ...
11
votes
1answer
394 views
Factorial canceling on expansion of binomial coefficients on Concrete Mathematics
On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as:
\[
\frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z}
\]
where
\[
...
4
votes
1answer
148 views
Evaluating a limit involving binomial coefficients.
If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer:
$$\lim_{n\rightarrow ...
4
votes
2answers
341 views
Limit of alternating sum with binomial coefficient
I need to find a limit, or approximation for $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(a+bk)$ for, say, an $a,b\in (0,10)$. It is not so important what values $a$ and $b$ have. It would be ...
5
votes
3answers
199 views
A question about limit
My question is: What is the result of this limit:
$\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?
21
votes
6answers
947 views
Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$
Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
