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 ,....
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
$\displaystyle \chi$ $\displaystyle \sigma$