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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 hard-and-fast rules, running down the following steps (in order) may be helpful.
Strategy to test series
- If you see that the terms $a_n$ do not go to zero, you know the series diverges by the Divergence Test.
- If a series is a $p$-series, with terms $\frac{1}{n^p}$, we know it converges if $p>1$ and diverges otherwise.
- If a series is a geometric series, with terms $ar^n$, we know it converges if $|r|<1$ and diverges otherwise. In addition, if it converges and the series starts with $n=0$ we know its value is $\frac{a}{1-r}$. (If it starts with another value of $n$, some work must be done to determine its value.)
- If a series is similar to a $p$-series or a geometric series, you should consider a Comparison Test or a Limit Comparison Test. These test only work with positive term series, but if your series has both positive and negative terms you can test $\sum|a_n|$ for absolute convergence.
- If the series has alternating signs, the Alternating Series Test is helpful; in particular, in a previous step you have already determined that your terms go to zero. However, the AST will not indicate whether a series converges absolutely or conditionally - determining this will require other tests.
- If your terms contain factorials, or factorials and $n^{th}$ powers, the Ratio Test might be helpful. This test does not care if your terms are negative, and may determine absolute convergence of the series. However, this test will fail for $p$-series and all rational functions of $n$, so don't try the Ratio Test on these.
- If your terms contain $n^{th}$ powers, the Root Test may be helpful. (If you have a geometric series, you will already know it before coming to this step.) This test does not care if your terms are negative, and may determine Absolute Convergence of the series.
- If your terms are positive and decreasing, and easily integrated (when viewed as $f(x)$ where $f(n)=a_n$), the Integral Test may be helpful.
A review of all series tests
Consider the series $\displaystyle\sum_{n}^\infty a_n$.
Integral Test: If $a_n = f(n)$, where $f(x)$ is a non-negative non-increasing 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 positive-term series.
If $a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.If $b_n \le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
Limit comparison Test: If $\sum a_n$ and $\sum b_n$ are positive-term 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.
If $L > 1$, or the limit goes to $\infty$, then $\sum a_n$ diverges.
If $L=1$ or if $L$ does not exist, then the test fails, and we know nothing.
Root Test: Let $L = \displaystyle\lim_{n \to \infty}\sqrt[n]{|a_n|}$.
If $L<1$, then $\sum a_n$ converges absolutely.
If $L>1$, or the limit goes to infinity, then $\sum a_n$ diverges.
If $L=1$, or if $L$ does not exist, then the test fails, and we know nothing.
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