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