Science:Infinite Series Module/Units/Unit 2/2.4 The Ratio Test/2.4.04 A Simple Ratio Test Example

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Using only the ratio test, determine whether or not the recursive sequence

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} a_1 = 1, \qquad a_{k+1} = \frac{0.1 + \cos(k)}{\sqrt{k}}a_k, \quad k = 1, 2, 3, \ldots  \end{align} }

converges or diverges.

Complete Solution

Applying the ratio test yields

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \lim_{k \rightarrow \infty} \Big| \frac{a_{k+1}}{a_k} \Big| &= \lim_{k \rightarrow \infty} \Bigg| \frac{\frac{0.1 + \cos(k)}{\sqrt{k}}a_k}{a_k} \Bigg| && (1) \\ &= \lim_{k \rightarrow \infty} \Bigg| \frac{0.1 + \cos(k)}{\sqrt{k}} \Bigg| && (2)\\ \end{align}}



Since the limit equals , the ratio test tells us that the series converges.

Explanation of Each Step

Step (1)

To apply the ratio test, we must evaluate the limit

In our problem, we can use

and substitute this into our limit.

Step (2)

In Step (2), we only cancel the in the numerator and denominator.

Step (3)

First observe that

Dividing everything by the square root of we obtain

Step (4)

In Step (4) we only evaluate the limit:

which equals zero because the numerator is a constant and the denominator goes to infinity.

Step (5)

In Step (5) we apply the Squeeze Theorem.

Potential Challenge Areas

Getting Started

Because the question asks us to apply the ratio test, we know that we will start our solution by using the formula

Recursive Formula

Most problems involving convergence tests don't involve recursive formulas. But with the ratio test, we apply

and use the given recursion equation for . In our case, our recursion equation is

which we substitute into the numerator, allowing us to cancel the in the numerator and denominator. This trick is a bit harder to apply for the other convergence tests.