Showing that this integral diverges

Question

I need to compute: $$ \lim_{n \to \infty} \int_{0}^{\infty} (1+x)^{np} \prod_{j=1}^{p} (1+xt_j)^{-n}dx $$ Where the the $t_j \in [0,1] \; \; \forall j$

How can I mathematically show that is is equal to $\infty$?

Answer 1

just majorate $$ 1\leq 1+xt_{j}\leq 1+x\Rightarrow (1+x)^{-n}\leq( 1+xt_{j})^{-n}\leq 1\\\prod_{1}^{p}(1+x)^{-n}\leq\prod_{1}^{p}( 1+xt_{j})^{-n}\\then :\lim_{\infty }\int_{0}^{\infty }(1+x)^{np}(1+x)^{-np}=\infty $$

Answer 2

$$\left(\frac{1+x}{1+t_j x} \right)^n \geqslant \left(\frac{1+x}{1+ x} \right)^n =1$$ Hence $$f(x)=\left(1+x \right)^{np}\prod_{j=1}^{p} (1+t_j x)^{-n} =\prod_{j=1}^p\left(\frac{1+x}{1+t_j x} \right)^n\geqslant 1$$ So $$\int_0^{\infty} f(x) dx \geqslant\int_0^{\infty} 1 dx =\infty$$