1. ## Convergence proof(integral)

I have a problem with proving that the next integral is converges:

$\int^0_{-1}\frac{ ln(1+t)}{t}dt$

Thanks!

2. You can start from the general result [which of course can be demostrated]...

$\displaystyle \int_{0}^{1} \tau^{n}\ \ln \tau\ d\tau = -\frac{1}{(n+1)^{2}}$ (1)

... and then performing the substitution $1+t=\tau$ You obtain...

$\displaystyle \int_{-1}^{0} \frac{\ln (1+t)}{t}\ dt = - \int_{0}^{1} \frac{\ln \tau}{1-\tau}\ d\tau = - \sum_{n=0}^{\infty} \int_{0}^{1} \tau^{n}\ \ln \tau\ d\tau =$

$\displaystyle = \sum_{n=0}^{\infty} \frac{1}{(n+1)^{2}} = \frac{\pi ^ {2}}{6}$ (2)

Kind regards

$\chi$ $\sigma$

3. ## Oh!

Great! (This result I need to prove later actually)

I have tried to do something like this :

$\int^0_{-1}\frac{ln(1+t))}{t}dt<-2\int^1_0 ln(1+t)dt= -2[(t+1)ln(t+1)-(t+1)]^1_0<\infty$
Now, by comparison test...

Is that prove the convergence?

Thank you again!