Question

determine whether the sequence is increasing,decreasing or bounded, ln(n+2)/n+2

Answer 1

For answering the question we consider the function :
\[f(x)=\frac{\ln(x+2)}{x+2}\quad x\geq0.\]
We derive it :
\[f'(x)=\frac{\frac1{x+2}\times(x+2)-\ln(x+2)}{(x+2)^2}=\frac{1-\ln(x+2)}{(x+2)^2}
\]
And we have :
\[1-\ln(x+2)=0\iff \ln(x+2)=1\iff x+2=e\iff x=e-2\simeq=0.7\]
So we can sat that :
\[\forall x\geq 1,\quad f'(x)<0\]
We have :
\[f(1)=\frac{\ln 3}3\\
\lim_{x\to+\infty}f(x)=0
\]
So :
\[1) ~~f~\text{ is decreasing over }[1,+\infty(\\
2)~~\forall x\geq1,\quad 0\leq f(x)\leq \frac{\ln3}3.
\]
Now, our progression is defined by :
\[u_n=\frac{\ln(n+2)}{n+2}=f(n)\]
So :
\[1) \left(u_n\right)_{n\in{\mathbb N}}\text{ is decreasing }\\
2) \left(u_n\right)_{n\in{\mathbb N}}\text{ is bounded }
\]

Answer 2

Thanks a million.let me add for u,would u help me pls?

Answer 3

using definition show that lim n/2n-3 = 1/2 as as n approches to infinity