Question Number 134571 by liberty last updated on 05/Mar/21
$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{sequence}\:\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{ln}\:\mathrm{n}\right)^{\mathrm{100}} } \\ $$$$\mathrm{divergent}\:\mathrm{or}\:\mathrm{convergent}\: \\ $$
Answered by mathmax by abdo last updated on 05/Mar/21
![the function f(x)=(1/((lnx)^(100) )) is decreazing on]2,+∞[ so Σf(n) and ∫_2 ^∞ f(x)dx have the same nature but ∫_2 ^∞ f(x)dx=∫_2 ^∞ (dx/((lnx)^(100) ))=_(x=e^t ) ∫_(ln2) ^∞ ((e^t dt)/t^(100) ) we have lim_(t→∞) (e^t /t^(100) ) =+∞ ⇒this integral diverge ⇒this serie diverges](https://www.tinkutara.com/question/Q134574.png)
$$\left.\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{lnx}\right)^{\mathrm{100}} }\:\mathrm{is}\:\mathrm{decreazing}\:\mathrm{on}\right]\mathrm{2},+\infty\left[\:\mathrm{so}\:\Sigma\mathrm{f}\left(\mathrm{n}\right)\:\mathrm{and}\right. \\ $$$$\int_{\mathrm{2}} ^{\infty} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{nature}\:\:\mathrm{but} \\ $$$$\int_{\mathrm{2}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\int_{\mathrm{2}} ^{\infty} \frac{\mathrm{dx}}{\left(\mathrm{lnx}\right)^{\mathrm{100}} }=_{\mathrm{x}=\mathrm{e}^{\mathrm{t}} } \:\:\int_{\mathrm{ln2}} ^{\infty} \:\:\frac{\mathrm{e}^{\mathrm{t}} \mathrm{dt}}{\mathrm{t}^{\mathrm{100}} }\:\:\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{lim}_{\mathrm{t}\rightarrow\infty} \:\frac{\mathrm{e}^{\mathrm{t}} }{\mathrm{t}^{\mathrm{100}} }\:=+\infty\:\Rightarrow\mathrm{this}\:\mathrm{integral}\:\mathrm{diverge}\:\Rightarrow\mathrm{this}\:\mathrm{serie}\:\mathrm{diverges} \\ $$