Factorial function/Increasing/10! geq e^(11)/Estimate with exponential function/Exercise

a) Prove that for ${\displaystyle {}x\geq 1}$ the estimate

${\displaystyle {}\int _{1}^{\infty }t^{x}e^{-t}dt\leq 1\,}$

holds.

b) Prove that the function ${\displaystyle {}H(x)}$ defined by

${\displaystyle {}H(x)=\int _{1}^{\infty }t^{x}e^{-t}dt\,}$

for ${\displaystyle {}x\geq 1}$ is increasing.

c) Prove that ${\displaystyle {}10!\geq e^{11}+1}$.

d) Prove that for the factorial function for ${\displaystyle {}x\geq 10}$ the estimate

${\displaystyle {}\operatorname {Fac} \,x\geq e^{x}\,}$
holds.