Inconclusive convergence tests and Divergence Test

Question

What happens when the series is alternating and the tests that I know are inconclusive?

$\sum_{n=1}^\infty (-1)^n (n/(n+5))$.

The $b_n$ series is not decreasing, so I can't use the alternating series test (I took the derivative of $f(x)=x/(x+5)$ - slope is positive everywhere).

When I do the ratio test I get $=1$, so inconclusive.

So - am I allowed to simply ignore the alternating portion of the series and apply the divergence test which allows me to say lim as $n$ goes to $\infty$ of $a_n=1$ and since the limit does NOT $=0$, then the series diverges?

I thought I could only use the divergence test on non-alternating series!

Answer 1

You know that for a real or complex sequence $ ( u_n)$, we have the equivalence

$$\boxed{\lim_{n\to+\infty}u_n=0\iff \lim_{n\to+\infty}|u_n|=0}$$

So,

$$\lim_{n\to+\infty}|u_n|\ne 0\implies$$ $$\lim_{n\to+\infty}u_n \ne 0 \implies$$ $$\sum u_n \text{ is divergent}$$

Answer 2

Since the limit $\lim_{n\to\infty}(-1^n)\frac n{n+5}$ does not exist, then, in particular, you don't have $\lim_{n\to\infty}(-1^n)\frac n{n+5}=0$, and therefore the series diverges.