Linked Questions

4
votes
1answer
177 views

Is the sequences$\{S_n\}$ convergent? [duplicate]

Let $$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$ Is the sequences$\{S_n\}$ convergent? The following is my answer,but this is not correct. please give some hints. For all $x\in\mathbb{R}$, $$\lim_{n\...
208
votes
9answers
17k views

Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. I ...
6
votes
1answer
897 views

How to compute $\lim_{n\rightarrow\infty}e^{-n}\left(1+n+\frac{n^2}{2!}\cdots+\frac{n^n}{n!}\right)$ [duplicate]

There is a probabilistic method to solve it. But I am not familiar with probability. I am trying to compute it by analytic method, such as using L Hospital's rule or Stolz formula, but they are not ...
6
votes
2answers
4k views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
4
votes
2answers
2k views

Probability of cycle in random graph

I create a random directed graph, with N vertices and N edges, in the following process: A. Each vertex has a single outgoing edge. B. The target of that edge is selected at random from all N ...
3
votes
1answer
899 views

Upper bound on the partial sum of exponential series

How can I upper-bound the following function: $$f(n;a)=\sum_{k=0}^{n-1}\frac{(n-a\sqrt{n})^k}{k!}$$ where $0<a<\sqrt{n}$ is a constant. Since it's a partial sum of exponential series, a ...
3
votes
1answer
182 views

In the limit of $N \rightarrow \infty$, find solution $z$ to $\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$

Fix an integer $N$, and consider the unique positive solution $z$ to the following equation: $$\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$$ For $N = 0$, we find that $z ...
1
vote
1answer
160 views

Hypothesis testing with Poisson RV

Hi I am still having some trouble with the following question: I have mostly figured out the first part but after that is where I get confused Say we have random variables $X $~$ Poisson(\lambda)$ ...
1
vote
1answer
103 views

Asymptotics of partial exponential sum $ \sum\limits_{k=0}^{a n} \frac{n^k}{k!}$

The question Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ has been asked many times here, and there are different approaches for answering it. Motivated by this ...
4
votes
2answers
60 views

Asymptotic Gilbert-Varshamov Bound Using Hilbert's Entropy Formula

I am reading Walker's book Codes and Curves and am having trouble proving this Lemma regarding the Asymptotic Gilbert-Varshamov bound. Suppose that $q$ is a prime power and we define \begin{align*} ...