# Does the alternating series converge?

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
• @T.Bongers The term inside the floor function goes to $8/7$ which means that the limit will be $\ln 1 = 0$ – Ant May 10 '16 at 21:32
• @Ant How can that be possible? The limit is $\;\log\frac87\neq\log1=0\;$, isn't it? – DonAntonio May 10 '16 at 21:33
• @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

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