# Does the alternating series converge: $\sum\limits_{n=1}^{\infty}(-1)^n\ln\left[\frac{8n+2}{7n+1}\right]$?

I'm trying to find out whether the series $$\sum\limits_{n=1}^{\infty}(-1)^n\ln\left[\frac{8n+2}{7n+1}\right]$$ converges or not, but the alternating series test seems not to apply. What other tests can I use? Does this series converges or diverges?

• The terms don't even tend to zero, so.....
– user296602
May 10 '16 at 21:31
• @Joanpemo The square brackets used this way usually indicate the floor function.. If they are just parenthesis indeed you're correct
– Ant
May 10 '16 at 21:33
• @Ant Those look like brackets to me, not $\lfloor$ and $\rfloor$.
– user296602
May 10 '16 at 21:33
• Those look to me as brackets, too...The floor function usually uses $\;\lfloor.\rfloor\;$, I think. May 10 '16 at 21:34
• yeah, those are usual brackets May 10 '16 at 21:35

$$\log\frac{8n+2}{7n+1}\xrightarrow[n\to\infty]{}\log\frac87\neq0$$
$$(-1)^n\log\frac{8n+2}{7n+1}\rlap{\;\;\;\;/}\xrightarrow[n\to\infty]{}0\implies\text{ the series cannot converge}$$