calculate-U-n-1-arctan-n-x-x-2-dx-

Question Number 67011 by mathmax by abdo last updated on 21/Aug/19

c a l c u l a t e U_{n} = \int_{1}^{+ \infty} \frac{a r c t a n (n [x])}{x^{2}} d x

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

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

= \sum_{k = 1}^{\infty} a r c t a n (n k) {\frac{1}{k} - \frac{1}{k + 1}} = \sum_{k = 1}^{\infty} \frac{a r c t a n (n k)}{k^{2} + k}

\dots b e c o n t i n u e d \dots .