Is the following series convergent or divergent. If convergent find the sum if possible.

Question

How do you find if this series is convergent or divergent?

$$\sum_{n=1}^{\infty} \dfrac{\ln(n^2+1)}{\ln(n^3+1)}$$

Is someone able to solve this with all the steps?

Answer 1

For a series to converge, recall that the $n^{th}$ terms needs to converge to $0$.

We have $a_n = \dfrac{\ln(n^2+1)}{\ln(n^3+1)} = \dfrac{\ln(n^2) + \ln(1+1/n^2)}{\ln(n^3) + \ln(1+1/n^3)} = \dfrac{2\ln(n) + \ln(1+1/n^2)}{3\ln(n) + \ln(1+1/n^3)}$. Hence, $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \dfrac{2\ln(n) + \ln(1+1/n^2)}{3\ln(n) + \ln(1+1/n^3)} = \dfrac23$$ Hence, conclude that ...