Hello all,
I need to prove that the sequence $\displaystyle c_n = 1+\frac{1}{2}+...+\frac{1}{n} - ln(n+5)$ is increasing. I did compute $\displaystyle c_{n+1}-c_{n}$ but i couldn't find the sign.
Help please
$\displaystyle c_{n+1}-c_{n}=\frac{1}{n+1}-\ln\left(\frac{n+5}{n+6}\right)=\frac{1}{n+1}-\ln\left(1-\frac{1}{n+6}\right)\geqslant\frac{1}{n+1}-\ln\left(1-\frac{1}{n+1}\right)$.
It remains to show that $\displaystyle x\geqslant\ln(1-x)$. To see this merely note that $\displaystyle f(x)=x-\ln(1-x)\implies f(0)=0,\text{ }f'(x)=1+\frac{1}{1-x}\geqslant 0$.