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

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

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

holds.

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

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

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

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

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

${}\operatorname {Fac} \,x\geq e^{x}\,$

holds.

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