The Ratio and Root Tests

$\frac{\left|a_{n+1}\right|}{\left|a_n\right|}>1$ , which would indicate that the series is no longer decreasing. On the other hand, if this limit is less than 1, the series converges absolutely.

The Ratio Test: Suppose that

$\displaystyle{\lim_{n\to\infty} \frac{\left|a_{n+1}\right|}{\left|a_n\right|}} = L.$

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 this test is inconclusive, and we must do more work. We say the Ratio Test fails if $L=1$

Notice that the Ratio Test considers the ratio of the absolute values of the terms. As you might expect, the Ratio Test thus gives us information about whether the series

$\sum a_n$ converges absolutely.

Warning: There are examples with

$L=1$ that converge absolutely, examples that converge conditionally, and examples that diverge. DO: Apply the Ratio Test to 1) the absolutely convergent series

$\sum\frac{1}{n^2}$ , 2) the conditionally convergent series

$\sum-\frac{1}{n}$ , and 3) the divergent series

$\sum\frac{1}{n}$ .
1) As

$n\to\infty$ ,

$\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{(n+1)^2}}{\frac{1}{n^2}}=\frac{n^2}{(n+1)^2}\longrightarrow 1$ .
2) As

$n\to\infty$ ,

$\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$ .
3) As

$n\to\infty$ ,

$\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$ .
We used other tests to determine the convergence/divergence of these series - the Ratio Test fails to help us with these series.

DO: The only way a series could be conditionally convergent is if the Ratio Test fails for that series. Why?

Review of simiplification

As you work through this module, you must be able to work with ratios of factorials as well as ratio of powers. Recall that

$n!=1\cdot 2\cdot 3\cdots(n-1)\cdot n$ . DO: Simplify

$\frac{(n+1)!}{n!}$ .

$\frac{(n+1)!}{n!}=\frac{1\cdot 2\cdot 3\cdots(n-1)\cdot n\cdot (n+1)}{1\cdot 2\cdot 3\cdots(n-1)\cdot n}=n+1$ after all the cancellation.

DO: Simplify

$\displaystyle\frac{\frac{50^{n+1}}{(n+1)!}}{\frac{50^n}{n!}}$

$\displaystyle\frac{\frac{50^{n+1}}{(n+1)!}}{\frac{50^n}{n!}}=\frac{50^{n+1}}{(n+1)!}\cdot\frac{n!}{50^n}=\frac{50^{n+1}}{50^n}\cdot\frac{n!}{(n+1)!}=50\cdot\frac{1}{n+1}=\frac{50}{n+1}$

A couple of worked out examples of the Ratio Test are contained in the video, as well as the ideas of why the Ratio Test works.

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

The Ratio Test

Review of simiplification