# Thread: Need a hint! (Proving convergence)

1. ## Need a hint! (Proving convergence)

I need a hint, the question is written in the picture, thank you.

2. ## Re: Need a hint! (Proving convergence)

see what the natural log of it converges to.

3. ## Re: Need a hint! (Proving convergence)

what do you mean by ''natural log of it''? thank you.

5. ## Re: Need a hint! (Proving convergence)

Originally Posted by davidciprut
I need a hint, the question is written in the picture, thank you.
Maybe you should see if have copied it correctly.

$\displaystyle {\lim _{n \to \infty }}{\left( {1 + \frac{1}{n}} \right)^{{n^2}}} = \infty$

$\displaystyle {\lim _{n \to \infty }}{\left( {1 + \frac{1}{n}} \right)^{{n}}} = e$

6. ## Re: Need a hint! (Proving convergence)

Wait , what? I was sure it converged to e. The difference with the second sequence you wrote was that it goes faster than the second, but it's still bounded and and converges to e. At least that's what I thought....

7. ## Re: Need a hint! (Proving convergence)

\displaystyle \begin{align*}\lim_{n \to \infty} \left(1+\dfrac{1}{n}\right)^{n^2} & = \lim_{n \to \infty} \exp\left(n^2\ln\left(1+\dfrac{1}{n}\right)\right) \\ & = \exp\left( \lim_{n \to \infty} \dfrac{\ln\left(1+\dfrac{1}{n}\right) }{n^{-2}} \right) \\ & \stackrel{0}{\stackrel{=}{0}} \exp\left( \lim_{n\to \infty} \dfrac{ \left( \dfrac{ -n^{-2} }{ 1+\dfrac{1}{n} } \right) }{ -2n^{-3} } \right) \\ & = \exp\left( \lim_{n \to \infty} \dfrac{n}{2\left(1+\dfrac{1}{n}\right)} \right) \\ & = e^\infty = \infty\end{align*}

Edit: I used L'Hospital's Rule on line 3, then for line 4, I multiplied top and bottom of the fraction by $\displaystyle -n^3$. Then, the limit as $\displaystyle n\to \infty$ of $\displaystyle \dfrac{n}{2\left(1+\dfrac{1}{n}\right)} = \infty$

8. ## Re: Need a hint! (Proving convergence)

Originally Posted by davidciprut
Wait , what? I was sure it converged to e. The difference with the second sequence you wrote was that it goes faster than the second, but it's still bounded and and converges to e. At least that's what I thought....
It is very to show $\displaystyle {\lim _{n \to \infty }}{\left( {1 + \frac{1}{n}} \right)^{{n^2}}} = \infty$.

If $\displaystyle 1<r<e$ then $\displaystyle \exists K\in\mathbb{Z}^+[r<\left(1+\frac{1}{K}\right)^K<e]$.

The e sequence is increasing so $\displaystyle \forall n>K$ we have $\displaystyle r^n<\left( {1 + \frac{1}{n}} \right)^{{n^2}}}$

$\displaystyle {\lim _{n \to \infty }}{r^n} = \infty$

9. ## Re: Need a hint! (Proving convergence)

Wow, I feel so stupid right now... I got it... Thank you all