let-U-n-0-arctan-nt-1-n-2-t-2-dt-with-n-natural-1-1-calculate-U-n-2-calculate-lim-n-n-2-U-n-3-study-the-convergence-of-U-n-

Question Number 62812 by mathmax by abdo last updated on 25/Jun/19

l e t U_{n} = \int_{0}^{+ \infty} \frac{a r c t a n (n t)}{1 + n^{2} t^{2}} d t w i t h n n a t u r a l ⩾ 1

1) c a l c u l a t e U_{n}

2) c a l c u l a t e {l i m}_{n \to + \infty} n^{2} U_{n}

3) s t u d y t h e c o n v e r g e n c e o f Σ U_{n}

Commented by mathmax by abdo last updated on 26/Jun/19

1) b y p a r t s u^{^{'}} = \frac{1}{1 + n^{2} t^{2}} a n d v = a r c t a n (n t) \Rightarrow

U_{n} = {[\frac{1}{n} {(a r c t a n (n t))}^{2}]}_{0}^{+ \infty} - \int_{0}^{\infty} \frac{1}{n} a r c t a n (n t) \frac{n}{1 + n^{2} t^{2}} d t

= \frac{π^{2}}{4 n} - \int_{0}^{\infty} \frac{a r c t a n (n t)}{1 + n^{2} t^{2}} d t = \frac{π^{2}}{4 n} - U_{n} \Rightarrow 2 U_{n} = \frac{π^{2}}{4 n} \Rightarrow U_{n} = \frac{π^{2}}{8 n}

2) w e h a v e n^{2} U_{n} = \frac{n π^{2}}{8} \Rightarrow {l i m}_{n \to + \infty} n^{2} U_{n} = + \infty

3) t h e n u m e r i c s e r i e Σ \frac{π^{2}}{8 n} d i v e r g e s \Rightarrow Σ U_{n} d i v e r g e s .