Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent

Question

I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that $\lim\limits_{x \to ∞} $$ x^{\frac{1}{x}}$ is 1 but I cant decipher how it useful. any help? THANKS

Answer 1

Notice that $$\frac1{x^{\frac1x+1}}\sim_\infty \frac1x$$ which means that: for $\epsilon>0$ there is $A>0$ such that for $x\ge A$ we have $$\frac1{x^{\frac1x+1}}\ge (1-\epsilon)\frac1x$$

From this inequality we conclude that the given integral is divergent since $$\int_A^\infty\frac{dx}{x}=+\infty$$