# Basic Alternating Sum Test clarification.

So, to use Alternating Sum Test, $b_n$ must be decreasing and its $\lim_{n\to \infty} = 0$.

What do we do when when one of the conditions fail? We try to take a limit/use another test?

Example:

$-\frac 25 + \frac46 - \frac67 + \frac88 - \frac {10}{9} + \ldots +$

That I think would be equal to: $\sum_{n=1}^\infty \frac{(-1)^n 2n} {4+n}$, which isn't decreasing and $\lim_{n\to \infty} b_n \neq 0$. $\lim_{n\to \infty} a_n$ then doesn't exist? Yet, it says that the sum is convergent. Where do I go wrong?

The sum is divergent by a more basic test: $$\hbox{if}\quad a_n\not\to0\quad\hbox{then}\quad\sum_{n=1}^\infty a_n\ \hbox{diverges}.$$ And in this case $$\frac{2n}{4+n}\to2$$ so $\frac{(-1)^n2n}{4+n}$ does not tend to zero.
• Thanks but that's not what I was asking for. My question is, how can the book say that it's convergent when $\lim b_n \neq 0$? – latenight_help Apr 30 '14 at 22:52