Is there any particular method to use when dealing with these? Let me explain my problem. For example:

Here are the rules we know:

1. The Alternating Series TestIf the series

$\displaystyle

\sum\limits_{n = 1}^\infty {( - 1)^{n + 1} u_n = u_1 - u_2 + u_3 - u_4 + \cdot \cdot \cdot }

$

converges if all three of the following conditions are satisfied:

a. $\displaystyle

u_n

$ is positive for alln.

b. $\displaystyle

u_n \geqslant u_{n + 1}

$ for all $\displaystyle

n \geqslant N

$, for some integer N.

c. $\displaystyle

u_n \to 0

$

2. Absolute ConvergenceGiven $\displaystyle

\sum\limits_{n = 1}^\infty {a_n }

$ if $\displaystyle

\sum\limits_{n = 1}^\infty {|a_n } |

$ converges, we say that $\displaystyle

\sum\limits_{n = 1}^\infty {a_n }

$ converges absolutely.

3. Conditional ConvergenceIf $\displaystyle

\sum\limits_{n = 1}^\infty {|a_n } |

$ diverges, but $\displaystyle

\sum\limits_{n = 1}^\infty {a_n }

$ converges, then we say that $\displaystyle

\sum\limits_{n = 1}^\infty {a_n }

$ converges conditionally.

Example One:

$\displaystyle

\sum\limits_{n = 1}^\infty {( - 1)^{n + 1} \frac{{3 + n}}

{{5 + n}}}

$

So where do I start with an alternating series? In this instance, we know $\displaystyle

{\frac{{3 + n}}

{{5 + n}}}

$ is an increasing function, so it fails part b of the alternating series test. Also $\displaystyle

{\mathop {\lim }\limits_{n \to \infty } \frac{{3 + n}}

{{5 + n}}}

$ approaches one as n goes to infinity. So this alternating series fails the alternating series test completely. So therefore it diverges?

Example Two

$\displaystyle

\sum\limits_{n = 1}^\infty {( - 1)^{n + 1} } \frac{n}

{{n^3 + 1}}

$

What about this one? I think this one meets of the conditions of the alternating series test. So then, if a series meets all the conditions, do I then use the conditional and absolute convergence tests? or do I just leave it at "converges by alternating series test" What is a good rule of thumb for these series?