Amazing limit

Let $(x_n)_{n=1}^{\infty} \text{ and } (y_n)_{n=1}^{\infty}$ be two real sequences defined as

$x_i = \displaystyle \sum_{i=1}^{n} \dfrac{1}{3i-2} \text{ and } y_i = \ln(3i-2)$

Now, $(y_n)_{n=1}^{\infty}$ is a strictly monotonic increasing function, and it diverges to $\infty$ .

Therefore, by Stolz-Cesáro Theorem , we have:

$\lim_{n \to \infty} \frac{x_n}{y_n} = \lim_{n \to \infty} \frac{x_{n+1} - x_n}{y_{n+1} - y_n}$

$=\lim_{n \to \infty} \dfrac{\frac{1}{3n+1}}{\ln(3n+1) - \ln(3n-2)}$

$=\lim_{n \to \infty} \dfrac{1}{(3n+1)\ln(1+\frac{3}{3n-2})}$

$=\lim_{n \to \infty} \dfrac{3n-2}{3(3n+1)} = \boxed{\dfrac{1}{3}}$

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The last step is obtained from the pen-ultimate step as we know that $\lim_{x \to 0} \ln(1+x) = x$