Prove convergence or divergence: Integral Test

Question

How can I prove that the series $$\sum_{n=1} \frac{1}{(n+1)(\ln(n)+1)}$$ converges or diverges. I have tried the ratio test, which resulted in

$\lim\limits_{x \to ∞} \frac{\frac{1}{((n+1)+1)(ln(n+1)+1)}}{\frac{1}{(n+1)(ln(n)+1)}}≈ \frac{nln(n)}{nln(n+1)}=1 $

I have attempted to use the integral test, and while [I'm fairly confident] the:

• derivative is always negative, so the function is always decreasing
• function is always positive
• function is continuous

I'm not sure how to compute the integral. Again, the integral test might not even be the right thing to use in this circumstance; if it's not, I'm not sure what other tests to use.

To contextualize my knowledge, I'm only in Calculus BC, and I don't know much abstract math or anything, so please no complex solutions; thank you.

Answer 1

Diverges by the limit comparison if one compares the given series with the divergent series $\sum_{n = 2}^{\infty} \frac{1}{n \ln n } $ which is evident as one can see from the integral test $\int_2^{\infty} \frac{ dx }{ x \ln x } = \infty $

Answer 2

$\int_1^{\infty} \frac 1 {(x+1)(\ln\, x+1)}\, dx=\int_0^{\infty} \frac {e^{y}} {(e^{y}+1) (y+1)}\, dy$. If you use the fact that $\frac {e^{y}} {e^{y}+1} \geq \frac 1 2$ you see easily that the integral diverges.