Determine $\sum^{\infty}_{n=1} \frac{(-1)^{n-1}}{\ln(n+4)}$ Determine $$\sum^{\infty}_{n=1} \frac{(-1)^{n-1}}{\ln(n+4)}$$
I want to use the alternating series test, and I have already computed the limit as $n\to\infty = 0$. Now I have to show the the series is decreasing. How can I do this? 
In an example I found online, they attempted to show that the series $\sum^{\infty}_{n=1} \dfrac{(-1)^{n+1}\ln(n)}{n}$ is decreasing, and how they did that was by taking the derivative of $\dfrac{\ln(n)}{n}$, and showing that it is negative. Is this correct. Why did they leave out the $(-1)^{n+1}$?

Yes, what they did is correct. You leave the $-1$ out because you know exactly how that term behaves... you need to know how the rest of the term behaves without the annoyance of an oscillating term.
