Claim: Let $f:[a,b) \to R $ be a non-negative and continuous function.
Suppose $\int_{a}^{b} e \ ^{f(t)} dt $ converge , than $\int_{a}^{b} f(t) \ ^ 7dt$ converge too.
I think this claim is true.
When i tried to prove it I showed that $lim_{x \to \infty} \dfrac{x \ ^ 7}{e \ ^ x} = 0$ than i wanted to say that this implies that $lim_{x \to \infty} \dfrac{f(x) \ ^ 7}{e \ ^ {f(x)}} = 0$ , and than to finish the proof by comparison test. but this is true only if $lim_{x \to \infty} f(x) = \infty $ which is not necessarily true.
How could I continue from here or is there another way to prove this claim ?
Thanks !
If we examine Taylor series $$e^{f(t)}=1+\frac{f(t)}{1!}+\frac{f^2(t)}{2!}+... \ge \frac{f^7(t)}{7!} $$
So by comparison test, we know the following converges. $$\int_a^b\frac{f^7(t)}{7!}dt$$