The Ratio and Root Tests

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The Ratio Test

Let $\sum a_n$ be a series. The Ratio Test involves looking at $$\displaystyle{\lim_{n \to \infty} \frac{\left|a_{n+1}\right|}{\left|a_n\right|}}$$ to see how a series behaves in the long run. As $n$ goes to infinity, this ratio measures how much smaller the value of $a_{n+1}$ is, as compared to the previous term $a_n$, to see how much the terms are decreasing (in absolute value). If this limit is greater than 1, then for all values of $n$ past a certain point, the ratio $\frac{\left|a_{n+1}\right|}{\left|a_n\right|}>1$, 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.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

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

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

The Ratio Test

Review of simiplification