Linked Questions

6
votes
8answers
335 views

Evaluate $\lim_{n \rightarrow \infty} \frac {[(n+1)(n+2)\cdots(n+n)]^{1/n}}{n}$ [duplicate]

Evaluate $$\lim_{n \rightarrow \infty~} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ Attempt: Let $$y=\lim_{n \rightarrow \infty} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ $$\...
9
votes
9answers
193 views

limit of product of $(a_1a_2.\dots a_n)^{\frac{1}{n}}$ [duplicate]

How to calculate the following limit $$ \lim_{n\rightarrow \infty} \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots \left(1+\frac{n}{n}\right) \right]^\frac{1}{n} .$$ I was trying ...
3
votes
5answers
138 views

$\lim_{n\to \infty} {1\over n}\sqrt[n]{(n+1)(n+2)\cdots(2n)}$ [duplicate]

$$\lim_{n\to \infty} {1\over n}\sqrt[n]{(n+1)(n+2)\cdots(2n)}$$ My attempt: \begin{align} \lim_{n\to \infty} {1/ n}\sqrt[n]{(n+1)(n+2)\cdots(2n)} &= \lim_{n\to \infty} \sqrt[n]{(1+{1/ n})(1+{2/ ...
5
votes
2answers
143 views

What is the $\lim_{n\to \infty} {\sqrt[n]{(1+1/n)(1+2/n)…(1+n/n)}}$ [duplicate]

Because $(1+\frac{k}{n})\leq (1+\frac{1}{n})^k$ $$ {\sqrt[n]{(1+1/n)(1+2/n)...(1+n/n)}}\leq\sqrt[n]{(1+\frac{1}{n})^{n(n+1)/2}}=(1+\frac{1}{n})^{(n+1)/2}$$ and so $$\lim_{n\to \infty}{\sqrt[n]{(1+1/n)(...
0
votes
3answers
138 views

Find the limit of the following sequence. (Real Analysis) [duplicate]

Find the limit of the following sequence. $$a_n = \frac{\sqrt[n]{(n+1)(n+2)...(2n)}}{n}$$ I tried couple of methods: Stolz, Squeeze, D'Alambert. But I can not seem to make a conclusion on the limit.
2
votes
3answers
83 views

What value does $\frac{1}{n} \sqrt[n]{n(n+1)(n+2)\cdots(2n)}$ tend to? [duplicate]

I need to find where this sequence tends to: $$\frac{1}{n} \sqrt[n]{n(n+1)(n+2)...(2n)}$$ My answer is $2$, by using a trick for the $n^{th}$ root: I take the inside and find where $\dfrac{A_{n+1}}{...
1
vote
4answers
133 views

Computing $\lim\limits_{n\to \infty}\frac1n\left((n+1)(n+2)…(n+n)\right)^{\frac1n}$ [duplicate]

The question asks to find the following limit. $$\lim_{n\to \infty}\left[\frac{\left((n+1)(n+2)...(n+n)\right)^{1/n}}{n}\right]$$ I tried taking $\log$ but I do not think that we get an L' Hopital ...
1
vote
2answers
106 views

Evaluate $\lim_{n\rightarrow\infty}$ $\left[\frac{\left(n+1\right)\left(n+2\right)…\left(n+n\right)}{n^{n}}\right]^{\frac{1}{n}}$ [duplicate]

Question: Evaluate lim$_{n\rightarrow\infty}$$\left[\frac{\left(n+1\right)\left(n+2\right).....\left(n+n\right)}{n^{n}}\right]^{\frac{1}{n}}$ My Approach Let $a_{n}=\frac{\left(n+1\right)\left(n+2\...
2
votes
1answer
76 views

How to calculate $\lim_{n \to \infty} \frac{1}{n}\sqrt[n] {(n+1)(n+2)…(n+n)}$? [duplicate]

How can I solve this limit: $\lim_{n \to \infty} \frac{1}{n}\sqrt[n] {(n+1)(n+2)...(n+n)}$? I've tried to do it by Sandwich, but I only obtained this: $\frac{1}{n}\sqrt[n] {(n+1)^n} \leq \frac{1}{...
39
votes
26answers
2k views

What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of $...
90
votes
7answers
8k views

Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
6
votes
5answers
9k views

Calculating the limit of $[(2n)!/(n!)^2]^{1/n}$ as $n$ tends to $\infty$

Analysis textbook by Shanti Narayan, is asking to prove the limit $\lim {\left({\dfrac{(2n)!}{(n!)^2}}\right)}^{1/n} \to \frac{1}{4}$ as $n \to \infty$. I tried but was unable to find the solution. ...
12
votes
4answers
393 views

How to compute $\lim_{n\rightarrow\infty}\frac1n\left\{(2n+1)(2n+2)\cdots(2n+n)\right\}^{1/n}$

If $\displaystyle f(n)=\frac1n\Big\{(2n+1)(2n+2)\cdots(2n+n)\Big\}^{1/n}$, then $\lim\limits_{n\to\infty}f(n)$ equals: $$ \begin{array}{} (\mathrm{A})\ \frac4e\qquad&(\mathrm{B})\ \frac{27}{4e}\...
13
votes
2answers
696 views

To evaluate limit of sequence $\left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$

How do I evaluate the limit of the following sequence $$a_n = \left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$$ I tried to take log and ...
14
votes
2answers
3k views

Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$

Today my lecturer put up on the board that: If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n>0$ then $\displaystyle \limsup\limits_{n\to\infty}\left(a_n^{\frac{1}{n}}\right)=\...

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