# Need a hint! (Proving convergence)

• Dec 20th 2013, 11:28 AM
davidciprut
Need a hint! (Proving convergence)
I need a hint, the question is written in the picture, thank you.
• Dec 20th 2013, 11:51 AM
romsek
Re: Need a hint! (Proving convergence)
see what the natural log of it converges to.
• Dec 20th 2013, 11:54 AM
davidciprut
Re: Need a hint! (Proving convergence)
what do you mean by ''natural log of it''? thank you.
• Dec 20th 2013, 11:59 AM
romsek
Re: Need a hint! (Proving convergence)
• Dec 20th 2013, 12:24 PM
Plato
Re: Need a hint! (Proving convergence)
Quote:

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.

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

$\displaystyle {\lim _{n \to \infty }}{\left( {1 + \frac{1}{n}} \right)^{{n}}} = e$
• Dec 20th 2013, 12:49 PM
davidciprut
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....
• Dec 20th 2013, 12:58 PM
SlipEternal
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 $\displaystyle \dfrac{n}{2\left(1+\dfrac{1}{n}\right)} = \infty$
• Dec 20th 2013, 01:07 PM
Plato
Re: Need a hint! (Proving convergence)
Quote:

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 $\displaystyle r^n<\left( {1 + \frac{1}{n}} \right)^{{n^2}}}$

$\displaystyle {\lim _{n \to \infty }}{r^n} = \infty$
• Dec 20th 2013, 01:17 PM
davidciprut
Re: Need a hint! (Proving convergence)
Wow, I feel so stupid right now... I got it... Thank you all