domain of convergence $\int_{1}^{+\infty}\frac{dt}{1+t^x}$

Question

let $x \in \mathbb{R} $,

how to determine the domain of convergence $\int_{1}^{+\infty}\frac{dt}{1+t^x}$ ,
my attempts :
I known if $x<0$ we get $\frac{1}{1+t^{x}}\rightarrow 1 [ t \rightarrow +\infty] $

Answer 1

Also divergent for x=0 ; If x>0 and t>=1 then $\frac{1}{(t^x)+ (t^x)}$ <= $\frac{1}{1+t^x}$ <= $\frac{1}{t^x}$ so that the convergence domain is the same as for $\frac{1}{t^x}$ which you can integrate directly to see that the convergence domain is the interval (1,$\infty$)

Answer 2

If $0<x\leq 1$ then $\int_1^{\infty} \frac 1 {1+t^{x}} \, dt \geq \int_1^{\infty} \frac 1 {t^{x}+t^{x}} \, dt=\infty$ as seen by direct computation. For $x>1$ $\,$ $\int_1^{\infty} \frac 1 {1+t^{x}} \, dt \leq \int_1^{\infty} \frac 1 {t^{x}} \, dt <\infty$ again by direct computation.