# Thread: Sequence and limit

1. ## Sequence and limit

Calculate limit of every sequence definite in N* by :
First sequence :

Second sequence :

2. $\displaystyle lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=\left(1+\fra c{1}{\infty}\right)^{\infty}$

you know that $\displaystyle \frac{1}{\infty}=0$

so the final thing will happen

$\displaystyle \left(1+0\right)^{\infty}=1$ there is a limit so it is converge

the second sequence is less than 2

my solution is wrong so look at whole thread ...........

3. Thank you not sure for the first question , more of detail on the second question

4. The first limit is the definition of e.

5. Originally Posted by Amer
$\displaystyle \color{red}lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=\left(1+\fra c{1}{\infty}\right)^{\infty}$

you know that $\displaystyle \color{red}\frac{1}{\infty}=0$

so the final thing will happen

$\displaystyle \color{red}\left(1+0\right)^{\infty}=1$ there is a limit so it is converge
That is completely wrong.

This is a well known limit: $\displaystyle \lim _{n \to \infty } \left( {1 + \frac{1} {n}} \right)^n = e$

6. The second one can be proved by induction after you make an educated guess about its limit.

7. Thanks Plato can you explain how it is equal e

8. That is a common definition of "e".

You can see a derivation, using a different definition here:http://www.physicsforums.com/showthread.php?t=176076