Prove-the-sequence-n-2-1-ln-n-100-divergent-or-convergent-

Question Number 134571 by liberty last updated on 05/Mar/21

Prove the sequence \underset{n = 2}{\sum^{\infty}} \frac{1}{{(\ln n)}^{100}}

divergent or convergent

Answered by mathmax by abdo last updated on 05/Mar/21

the function f (x) = \frac{1}{{(lnx)}^{100}} is decreazing on] 2, + \infty [so Σ f (n) and

\int_{2}^{\infty} f (x) dx have the same nature but

\int_{2}^{\infty} f (x) dx = \int_{2}^{\infty} \frac{dx}{{(lnx)}^{100}} =_{x = e^{t}} \int_{\ln 2}^{\infty} \frac{e^{t} dt}{t^{100}} we have

\lim_{t \to \infty} \frac{e^{t}}{t^{100}} = + \infty \Rightarrow this integral diverge \Rightarrow this serie diverges