Question

does the series ∑ 1/n⋅[1+(ln n)²] converge or diverge

n=1 below ∑

∞ on top of ∑

Answer 1

$\sum _{n=1}^{\infty \:}\frac{1}{n}\left(1+\left(\ln \left(n\right)\right)^2\right)$

$As \space a_n=\frac{1}{n}\left(1+\left(\ln \left(n\right)\right)^2\right) ; a_x=\frac{1}{x}\left(1+\left(\ln \left(x\right)\right)^2\right)$

$L= \int_1^\infin \frac{1}{x}\left(1+\left(\ln \left(x\right)\right)^2\right)=\ln \left|x\right|+\frac{1}{3}\ln ^3\left(x\right)+C|_1^\infin =\infin$

Hence $\sum _{n=1}^{\infty \:}\frac{1}{n}\left(1+\left(\ln \left(n\right)\right)^2\right)$ diverges