Linked Questions

5
votes
5answers
1k views

Prove that $a_n = \{\left(1+\frac{1}{n}\right)^n\}$ is bounded sequence, $ n\in\mathbb{N}$

How to prove the following: $a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$
4
votes
4answers
3k views

Prove $(1 + \frac{1}{n})^n$ is bounded above

I've checked similar questions on the site but couldn't find satisfactory solutions or hints. Also, is there a more general approach to proving whether a given sequence is bounded below or above?
6
votes
4answers
308 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( 1 ...
3
votes
2answers
222 views

Inequality $(1+\frac1k)^k \leq 3$

How can I elegantly show that: $(1 + \frac{1}{k})^k \leq 3$ For instance I could use the fact that this is an increasing function and then take $\lim_{ k\to \infty}$ and say that it equals $e$ and ...
1
vote
2answers
268 views

A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$ [closed]

Prove that $a$) the following sequence is increasing $$e_{n}=\left(1+\frac{1}{n}\right)^{n},\quad n\ge1;$$ $b$) the inequality below holds $$e_{n} \leq3,\quad n\ge1.$$