Trouble using alternating series test a. Find the domain of the definition $D \subset \mathbb R$ for the function
$$f(x)=\sum_{n=1}^\infty (-1)^n{x \over n+x}$$
I believe I should be using the alternating series test to show where it converges, but haven't used the test in a while. Looking for some direction on how to use it in this case. Thanks for any help :)
 A: As long as $x$ is not a negative integer, none of the terms have a zero in the denominator and eventually, when $n>-x$, the terms decrease monotonically to $0$. That is, if $n>-x$, then
$$
\frac{x}{n+1+x}\lt\frac{x}{n+x}
$$
and
$$
\lim_{n\to\infty}\frac{x}{n+x}=0
$$
Thus, the alternating series test says that the series converges if $x$ is not a negaitive integer.
A: If you take a real number which is not a negative integer , so that the denominator does not vanish, i.e. $D=\mathbb R \setminus \mathbb Z^-$  then you can use the alternating series test: 
If x positive then $\dfrac {x}{n+x}$ is positive and decreases monotonically to zero , therefore the test applies.
If x is negative then for large enough $n\in \mathbb N,$ say $n\geq n_0$ , $\dfrac{x}{n+x}$ is negative , therefore :$$\sum\limits_{n=1}^\infty (-1)^n\dfrac{x}{n+x}=\sum\limits_{n=n_0+1}^\infty (-1)^{n+1}\dfrac{|x|}{n+x}+\sum\limits_{n=1}^{n_0} (-1)^n\dfrac{x}{n+x}$$ 
and the alternating series test applies to the first (infinite ) series above.
A: Yes alternating series test works, for a fixed $x$. You have a series which is alternating in sign (for sufficiently large value of $n$, why? take $x=-10.5$ for example, for which value of $N$, does the sign start alternting after $n>N$?
$\dfrac{x}{n+x}\rightarrow 0$, except when ... ? so I think this works for any $x$ unless $x = ?$
