1. ## Convergence of integral

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

2. Originally Posted by miimi
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 $s\leqslant0$ this obviously diverges, so we may assume WLOG that $s>0$. So then notice that $f(x)=\frac{1}{x^{s(1+x)}}$ is positive, continuous, and non-increasing. Thus, we have by the integral test that the your integral shares convergence with $\sum_{n=1}^{\infty}\frac{1}{n^{s(1+n)}}$ so applying the root test we get $\lim_{n\to\infty}\frac{1}{\sqrt[n]{n^{s(1+n)}}}=\lim_{n\to\infty}\frac{1}{n^{s(1+\tf rac{1}{n})}}\leqslant\lim_{n\to\infty}\frac{1}{n^s }=0$..soo

3. Thanks for help, but i wrote it unclearly. The right version is:

∫1/(1+x)x^sdx
0

4. Originally Posted by miimi
Thanks for help, but i wrote it unclearly. The right version is:

∫1/(1+x)x^sdx
0
That's even easier. Once again, if $s\leqslant 0$ this diverges so assume that $s>0$ then $\frac{1}{(1+x)x^s}\underset{x\to\infty}{\sim}\frac {1}{x^{1+s}}$. Now, integrate the RHS and tell me if that converges.