Show that $$\int _ { 1 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t = O \left( \frac { 1 } { n 2 ^ { n } } \right) .$$ One may lower bound $1 + t ^ { 2 }$ by a polynomial of degree 1.
Show that
$$\int _ { 1 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t = O \left( \frac { 1 } { n 2 ^ { n } } \right) .$$
One may lower bound $1 + t ^ { 2 }$ by a polynomial of degree 1.