study-the-convergence-of-0-1-x-n-2-x-n-dx-n-N-

Question Number 66350 by mathmax by abdo last updated on 12/Aug/19

s t u d y t h e c o n v e r g e n c e o f \int_{0}^{\infty} (1 - \sqrt{\frac{x^{n}}{2 + x^{n}}}) d x n \in N

Commented by mathmax by abdo last updated on 14/Aug/19

l e t I = \int_{0}^{\infty} (1 - \sqrt{\frac{x^{n}}{2 + x^{n}}}) d x a n d f (x) = 1 - \sqrt{\frac{x^{n}}{2 + x^{n}}}

w e h a v e \sqrt{\frac{x^{n}}{2 + x^{n}}} = {(\frac{2 + x^{n} - 2}{2 + x^{n}})}^{\frac{1}{2}} = {(1 - \frac{2}{2 + x^{n}})}^{\frac{1}{2}}

\sim 1 - \frac{1}{2 + x^{n}} b u t \int_{0}^{\infty} (1 - \frac{1}{2 + x^{n}}) d x d i v e r g e s I d i v e r g e s .