Question Number 68244 by mathmax by abdo last updated on 07/Sep/19
![find nature of the serie Σ_(n=1) ^∞ arctan(n+(1/n))](https://www.tinkutara.com/question/Q68244.png)
$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{arctan}\left({n}+\frac{\mathrm{1}}{{n}}\right) \\ $$
Commented by mathmax by abdo last updated on 23/Sep/19
![let u_n =arctan(n+(1/n)) we have lim_(n→+∞) u_n =(π/2) ≠0 ⇒Σ u_n diverges](https://www.tinkutara.com/question/Q69410.png)
$${let}\:{u}_{{n}} ={arctan}\left({n}+\frac{\mathrm{1}}{{n}}\right)\:{we}\:{have}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} =\frac{\pi}{\mathrm{2}}\:\neq\mathrm{0}\:\Rightarrow\Sigma\:{u}_{{n}} {diverges} \\ $$