For which values of s is the integral
∞
∫(1/x^s(1+x)dx
0
a) absolutely convergent
b) conditionally convergent
c) divergent?
I know that
1
∫1/x^s dx is convergent <=> s<1 and
0
∞
∫1/x^s dx is convergent <=> s>1 but...
1
Note that if $\displaystyle s\leqslant0$ this obviously diverges, so we may assume WLOG that $\displaystyle s>0$. So then notice that $\displaystyle f(x)=\frac{1}{x^{s(1+x)}}$ is positive, continuous, and non-increasing. Thus, we have by the integral test that the your integral shares convergence with $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{s(1+n)}}$ so applying the root test we get $\displaystyle \lim_{n\to\infty}\frac{1}{\sqrt[n]{n^{s(1+n)}}}=\lim_{n\to\infty}\frac{1}{n^{s(1+\tf rac{1}{n})}}\leqslant\lim_{n\to\infty}\frac{1}{n^s }=0$..soo