1. ## limits of sequences

Calculate the limits of the sequences:

A)
Lim(n→+∞) nCos(n!)/(nē+1)

B)
Lim(n→+∞) f(n) , where:
f(1)=√2
f(2)=√2√2
f(3)=√2√2√2 ,....

2. For your a) question: note that $\displaystyle |\cos(n!)|\le 1$
Why not learn to post in symbols? You can use LaTeX tags.

3. Under the hypothesis that the second sequence is...

$\displaystyle a_{0}= 1$

$\displaystyle a_{1} = \sqrt{2}$

$\displaystyle a_{2} = \sqrt{2 \sqrt{2}}$

$\displaystyle \dots$

$\displaystyle a_{n+1} = \sqrt {2 a_{n}}$

$\displaystyle \dots$ (1)

... the difference equation that defines the sequence is...

$\displaystyle \Delta_{n} = a_{n+1}-a_{n} = \sqrt{2 a_{n}} - a_{n} = f(a_{n})$ (2)

The function $\displaystyle f(x)= \sqrt{2x} - x$ is illustrated here...

It has a single 'attractive fixed point' in $\displaystyle x_{0} = 2$ and because $\displaystyle \forall x>0$ is $\displaystyle |f(x)|<|2 - x|$, any 'initial value' $\displaystyle a_{0}>0$ will produce a sequence converging at 2 without oscillations...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$