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Strategy to Test Series and a Review of Tests
Notation: In this section, we will often use the following
series notations: 
$\displaystyle\sum_{n}^\infty
a_n=\sum_n a_n=\sum a_n$. 
$\left\{ \begin{array}{ll} &\text{All of these notations indicate that the index is $n$,}\\ &\text{but we aren't declaring where $n$ begins ($n=0$ or $n=1$ or $n=5$ etc.).}\\ \end{array}\right.$ 
As with techniques of integration, it is important to recognize
the form of a series in order
to decide your next steps. Although there are no
hardandfast rules, running down the following steps (in order)
may be helpful.
Consider the series $\displaystyle\sum_{n}^\infty a_n$.
Divergence Test: If
$\displaystyle\lim_{n \to \infty} a_n \ne 0$, then
$\displaystyle\sum_n a_n$ diverges. Integral Test: If $a_n = f(n)$, where $f(x)$ is a nonnegative nonincreasing function, then $\displaystyle\sum_{n}^\infty a_n$ converges if and only if the integral $\displaystyle\int_1^\infty f(x) \,dx$ converges. Comparison Test: This applies only to positiveterm series. If $a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. Limit comparison Test: If $\sum a_n$ and $\sum b_n$ are positiveterm series, and $\displaystyle\lim_{n \to \infty} \frac{a_n}{b_n} = L$, with $0<L<\infty$, then either $\sum a_n$ and $\sum b_n$ both converge or both diverge. Alternating Series Test: When our series is alternating, so that $\displaystyle\sum_n^\infty a_n=\sum_n^\infty(1)^nb_n$, if $b_n>0$, $\quad b_{n+1} \le b_n,\quad$ and $\quad\displaystyle\lim_{n \to \infty}b_n = 0$, then $\sum (1)^{n+1} b_n$ converges. Ratio Test: Let $L= \displaystyle{\lim_{n\to\infty} \frac{a_{n+1}}{a_n}}$. If $L < 1$, then $\sum a_n$ converges absolutely. Root Test: Let $L = \displaystyle\lim_{n \to \infty}\sqrt[n]{a_n}$. If $L<1$, then $\sum a_n$ converges absolutely.
