Strategies for Testing Series

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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 $\displaystyle\sum_{n}^\infty a_n$.

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 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.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Strategy to Test Series and a Review of Tests

Strategy to test series

A review of all series tests