let-give-I-n-0-dt-1-t-2-n-with-n-integr-and-n-1-1-prove-the-convergence-of-I-n-2-find-lim-n-I-n-3-study-the-convergence-of-the-serie-n-1-1-n-I-n-

Question Number 31966 by abdo imad last updated on 17/Mar/18

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

1) p r o v e t h e c o n v e r g e n c e o f I_{n}

2) f i n d {l i m}_{n \to \infty} I_{n}

3) s t u d y t h e c o n v e r g e n c e o f t h e s e r i e \sum_{n = 1}^{\infty} {(- 1)}^{n} I_{n} .