1. Sequence

Hi,

$\displaystyle (\forall n \in \mathbb{N})\ \ U_n=\frac{n^2}{2^n}$

I should calculate $\displaystyle \lim_{n\to +\infty} \frac{U_{n+1}}{U_n}$
and I must deduct that there is a $\displaystyle n_0$ in $\displaystyle \mathbb{N}$ such as :$\displaystyle (\forall n \ge n_0) \ \frac{U_{n+1}}{U_n} < \frac{3}{4}.$

For the limit, it's equal to $\displaystyle \frac{1}{2}$, but for the deduction, idon't know anything.

Thanks.

2. Originally Posted by lehder
Hi,

$\displaystyle (\forall n \in \mathbb{N})\ \ U_n=\frac{n^2}{2^n}$

I should calculate $\displaystyle \lim_{n\to +\infty} \frac{U_{n+1}}{U_n}$
and I must deduct that there is a $\displaystyle n_0$ in $\displaystyle \mathbb{N}$ such as :$\displaystyle (\forall n \ge n_0) \ \frac{U_{n+1}}{U_n} < \frac{3}{4}.$

For the limit, it's equal to $\displaystyle \frac{1}{2}$, but for the deduction, idon't know anything.

!!! The deduction is only the very application of the definition of limit!! For example, take $\displaystyle \epsilon=\frac{1}{8}$ ...