Ratio Test for Convergence

#10.8 Ratio Test for Convergence

Welcome back to AP Calculus BC! Indeed, there are many tests that you have learned from this unit, and we are headed to the final one, the ratio test, which is useful for series with exponentials or factorials. 👆

#🤔 What is the Ratio Test?

In the bigger picture, a ratio relates two numbers and is represented as a quotient. If there are 4 apples and 3 oranges in a basket, we can also say that the basket has a 4:3 apple-to-orange ratio. 🍎

The Ratio Test works as follows:

Like all the other tests, this theorem will make more sense as we move through example problems. Without further ado, let’s dive right in!

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#✏️ Ratio Test for Convergence Practice

Give them a try before you take a look at our walkthrough!

#1️⃣ Ratio Test Example 1

Determine whether the following series converges or diverges:

$$\sum \frac{5^n}{n!}$$

Recall that the Ratio Test deals with series $\sum a_n$. In this case, $a_n=\frac{5^n}{n!}$!

To complete our L = . . . equation above, we also need to find $a_{n+1}$. To get this, we simply replace n with (n + 1):

$$a_{n+1}=\frac{5^{n+1}}{(n+1)!}=\frac{5^n*5^1}{(n+1)*n!}$$

Now that we have both pieces of the puzzle, we can find $\lim\limits_{x→ \infty}\frac{a_{n+1}}{a_n}$.

\lim\limits_{n→ \infty}\frac{a_{n+1}}{a_n}=\lim\limits_{n→ \infty}\frac{a_{n+1}}{1}*\frac{1}{a_n}=\lim\limits_{n→ \infty}\frac{5^{n}*5}{(n+1)[object Object]\frac{n!}{5^n}

This cancels out $5^n$ and n!, and we’ll be left with…

$$\lim\limits_{n→ \infty}\frac{5}{n+1}=0$$

Plugging infinity into the denominator (per the limit) gives us 5 divided by a really, really large number, which gives us 0. Since 0 < 1, the series converges by the Ratio Test. ✅

#2️⃣ Ratio Test Example 2

Determine whether the following series converges or diverges:

$$\sum \frac{4^{2n+1}(n+1)}{(-10)^n}$$

First, we find that $a_n=\frac{4^{2n+1}(n+1)}{(-10)^n}$. We can further expand this term to get…

$$a_n=\frac{4^{2n+1}(n+1)}{(-10)^n}=\frac{16^n[object Object](n+1)}{(-10)^n}$$

Now, let’s find $a_{n+1}$ by plugging in (n + 1) into all the n terms in our equation:

a_{n+1}=\frac{4^{2(n+1)+1}((n+1)+1)}{(-10)^{(n+1)}}=\frac{4^{2n+3}(n+2)}{(-10)^{n+1}}=\frac{64[object Object](-10)}

Moving onto $\lim\limits_{x→ \infty}\frac{a_{n+1}}{a_n}$:

\lim\limits_{n→ \infty}\frac{a_{n+1}}{a_n}=\lim\limits_{n→ \infty}\frac{a_{n+1}}{1}[object Object]16^n(n+2)}{(-10)^n*(-10)}[object Object]4*(n+1)}

We can cancel out $(-10)^n$ and $16^n$. We’ll be left with:

$$\lim\limits_{n→ \infty}\frac{a_{n+1}}{a_n}=\lim\limits_{n→ \infty}\frac{16(n+2)}{-10(n+1)}=\lim\limits_{n→ \infty}\frac{16n+32}{-10n-10}$$

Applying L’hopital’s rule, we can further simplify this limit to…

$$\lim\limits_{n→ \infty}-\frac{16}{10}=-\frac{8}{5}$$

Remember, the theorem actually deals with the absolute value of the limit of $\frac{a_{n+1}}{a_n}$, so our final answer is actually 8/5. Since 8/5 > 1, the series diverges by the Ratio Test. ✅

#3️⃣ Ratio Test Example 3

Determine whether the following series converges or diverges:

$$\sum \frac{n+1}{n-2}$$

Like the previous examples, we find that $a_n=\frac{n+1}{n-2}$. Next, find $a_{n+1}$:

$$a_{n+1}=\frac{(n+1)+1}{(n+1)-2}=\frac{n+2}{n-1}$$

Now, we find $\lim\limits_{x→ \infty}\frac{a_{n+1}}{a_n}$. Applying L’hopital’s rule, we can further simplify this limit to…

$$\lim\limits_{n→ \infty}\frac{a_{n+1}}{a_n}=\lim\limits_{n→ \infty}\frac{a_{n+1}}{1}[object Object]\frac{n-2}{n+1})=1$$

This is a unique case! Recall from the Ratio Test that if L = 1, then the test is indeterminate. “Indeterminate” is a fancy way of saying we can’t determine whether the series above converges or diverges. That means we’ll have to use another test from our Unit 10 toolkit! ❓

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#🪐 Closing

This section wraps up all the tests (aka tools) you can use to determine whether a series converges, diverges, or remains inconclusive.

To briefly summarize, the tests we’ve encountered throughout the unit are:

• the nth term test (10.3)
• the integral test (10.4)
• the p-series test (includes harmonic series, 10.5)
• the direct comparison test (10.6)
• the limit comparison test (10.6)
• the alternating series test (10.7)
• the ratio test We’ve linked the corresponding study guides to each test just in case you need to review!

Once you become super familiar with these tests, you’ll find that you’ll only need one or two tests to determine [absolute or conditional] convergence or divergence. Tying these tests together might seem complicated and daunting at first especially when there are so many different tests to choose from, but with practice and pattern recognition skills, you’ll be golden! 🥇