Chapter 7: Q. 16 (page 656)

The Limit Comparison Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where L is ___, then either the series both converge or both diverge.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ ___, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ ___.
(There are two correct ways to fill in the last two blanks.)

Short Answer

Expert verified

Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms. If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where L is a finite positive number, then either the series both converge or both diverge.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ and role="math" localid="1649359537559" ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

Step by step solution

Step 1. Given Information

The given test is the limit comparison test.

Step 2. Explanation

If there are two series such that $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ then:
If $L = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ also converges.
If $L = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges , then ${鈭�}_{k = 1}^{鈭�} a_{k}$ also diverges.

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Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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