Science:Infinite Series Module/Units/Unit 2/2.4 The Ratio Test/2.4.05 Ratio Test Example with Exponent

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Question

Using only the ratio test, determine whether or not the series

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \sum_{k=1}^{\infty}   \frac{k}{5^k} \end{align} }

converges, diverges, or yields no conclusion.

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{(k+1) /5^{k+1}}{k/5^k} \Bigg| && (1) \\ &= \lim_{k \rightarrow \infty} \Bigg| \frac{k+1}{k} \cdot \frac{5^k}{5^{k+1}} \Bigg| && (2) \\ &= \lim_{k \rightarrow \infty} \Bigg( \Bigg| \frac{k+1}{k} \Bigg| \cdot \Bigg| \frac{5^k}{5^{k+1}} \Bigg| \Bigg) && (3) \\ &= \lim_{k \rightarrow \infty} \Bigg| \frac{k+1}{k} \Bigg| \cdot \lim_{k \rightarrow \infty}\Bigg| \frac{5^k}{5^{k+1}} \Bigg| && (4) \\ &= 1 \cdot \lim_{k \rightarrow \infty}\Bigg| \frac{5^k}{5^{k+1}} \Bigg| && (5) \\ &= \lim_{k \rightarrow \infty}\Bigg| \frac{5^k}{5 \cdot 5^k} \Bigg| && (6) \\ &= \lim_{k \rightarrow \infty}\Bigg| \frac{1}{5 } \Bigg| \\ &= \frac{1}{5}. \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 have

and we substitute them into our limit.

Step (2)

In Step (2), we use a little algebraic manipulation to make things easier to look at

Step (3)

Step (3) uses a property of absolute values. Recall that for real numbers and ,

Step (4)

Step (4) uses a property of limits values. Recall that for functions and , that

Step (5)

Here we evaluate a limit:

Step (6)

Some algebraic manipulation helps us see how we can simplify our problem. Recall that, using laws of exponentials, that for real numbers , that

so