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17calculus > infinite series > ratio test

Infinite Series - Ratio Test

The Ratio Test is probably the most important test and the test you will use the most as you are learning infinite series. It is used A LOT in power series. I believe it is the most powerful test of all and, if you look at the Infinite Series Table, you will see that it is listed first in Group 3. So I suggest you master it from the start. It's not hard, and if your algebra skills are strong, you might even find it fun to use. Also, the more familiar you are with it and the more practice you have, the sooner you will start to be able to look at a series and see almost right away if the Ratio Test will tell you what you need to know. Cool, eh?

Ratio Test

Let \( \sum{a_n} \) be a series with nonzero terms and let \( \displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} = L} \)

Three cases are possible depending on the value of L.
\( L < 1 \): The series converges absolutely.
\( L = 1 \): The Ratio Test is inconclusive.
\( L > 1 \): The series diverges.

Quick Details

used to prove convergence

yes

used to prove divergence

yes

can be inconclusive

yes

if \(L=\infty\) then \(L>1\) and the series diverges

\(a_n\) terms can be positive or negative or both

requires the use of limits at infinity

When To Use The Ratio Test

The ratio test is best used when you have certain elements in the sum. The best way to get a feel for this is to build a set of sheets containing examples of tests that work as you are working practice problems. This is one technique listed in the infinite series study techniques section. This is an extremely powerful technique that will help you really understand infinite series.
Here is a list of things to watch for.
1. Sums that include factorials.
2. Sums with exponents containing n.

How To Use The Ratio Test

In general, the idea is to set up the ratio \( \displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} = L} \) and evaluate it.
In detail, you need to determine what \(a_n\) is and then build \(a_{n+1}\), set up the fraction, combine like terms and then take the limit of each term. Setting up the limit and combining like terms are the easy parts. The challenge comes in taking the limit.
Key - It is important to remember to use the absolute value signs unless you are absolutely convinced that the term will always be positive. This is critical to practice up front since, once you get to Taylor Series, you can't and don't want to drop the absolute value signs. They are critical to the result. It is never wrong to include them and, as you work more problems, you will get a feel for when you need them and when you don't. In the practice problems and examples, we will use them unless we explicitly state that they are not needed. Some instructors are less rigid about this than others. As always, check with your instructor to see what they require.

Things To Notice
1. If you get \( \infty \) for the limit, this indicates divergence since it fits the case where the limit is greater than one. Notice that the theorem says nothing about the limit needing to be finite.
2. In the fraction that you are taking the limit of, the \( n+1 \) term is in the numerator and the \(n\) term is in the denominator. In order for the ratio test to work, they must appear like this. Do you know why? [answer ]
3. If you get one for the result, you cannot say anything about convergence or divergence of the series. You need to use another test. Sometimes, a comparison test ( either direct or limit ) will be the best next step.

Okay, it's time for some videos. This first video clip is a great overview of the ratio test. Notice that he doesn't use absolute value signs, so he requires that the terms be positive.

Dr Chris Tisdell: Series, Comparison + Ratio Tests

The beginning of this next video has a good discussion about the ratio test. Then the instructor shows two examples when the ratio test is inconclusive to emphasize that a series may converge or diverge when the ratio test is inconclusive.

MIT OpenCourseware: Ratio Test for Convergence

Okay, time for some practice problems.

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Practice Problems

Instructions - - Unless otherwise instructed, determine the convergence or divergence of the following series, using the ratio test, if possible. [These instructions imply that if the ratio test fails ( \(L = 1\) ), you need to use another test to prove convergence or divergence.]

Level A - Basic

Practice A01

\(\displaystyle{\sum_{n=0}^{\infty}{\left[\frac{n!}{2^n}\right]}}\)

answer

solution

Practice A02

\(\displaystyle{\sum_{n=0}^{\infty}{\left[\frac{n+2}{2n+9}\right]}}\)

answer

solution

Practice A03

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4^n}{n \cdot 3^n } \right] } }\)

answer

solution

Practice A04

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{10}}{10^n} \right] } }\)

answer

solution

Practice A05

\(\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \frac{2^n+5}{3^n} \right] } }\)

answer

solution

Practice A06

\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{3^k}{k^2} \right] } }\)

answer

solution

Practice A07

\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{2^k}{k!} \right] } }\)

answer

solution

Practice A08

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ n(5/4)^n \right] } }\)

answer

solution

Practice A09

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^n 2^{3n}}{(2n)!} \right] } }\)

answer

solution

Practice A10

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^3}{(\ln 3)^n} \right] } }\)

answer

solution

Practice A11

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^2}{2^n} \right] } }\)

answer

solution

Practice A12

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n(-3)^n}{4^{n-1}} \right] } }\)

answer

solution

Practice A13

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^{n+1} (n^2)2^n}{n!} \right] } }\)

answer

solution

Practice A14

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{7^n}{e^n} \right] } }\)

answer

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\sum_{n=2}^{\infty}{\left[\frac{n^2 2^n}{5^n}\right]}}\)

answer

solution

Practice B02

\(\displaystyle{\sum_{n=0}^{\infty}{\left[\frac{3n^2}{(2n-1)!}\right]}}\)

answer

solution

Practice B03

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{9^n}{(-3)^{n+1}n} \right] } }\)

answer

solution

Practice B04

\(\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \frac{3(-1)^n}{n^2+1} \right] } }\)

answer

solution

Practice B05

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(2n)!}{n!n!} \right] } }\)

answer

solution

Practice B06

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }\)

answer

solution

Practice B07

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4^n(n!)^2}{(n+2)!} \right] } }\)

answer

solution

Practice B08

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{n!} } }\)

solution


Level C - Advanced

Practice C01

\(\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \frac{(-5)^n}{4^{2n+1}(n+1)} \right] } }\)

answer

solution

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