Chapter 7: Q. 14 (page 631)

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?

Short Answer

Expert verified

$Hence \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L Where L is any positive real number then either both converges or diverges .$

Step by step solution

Step 1. Given information

$\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L$

Step 2. Limit comparison test

$The limit comparison test states that for and be two series with positive terms then If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L Where L is any positive real number then either both converges or diverges . If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = 0 and \sum_{k = 1}^{\infty} b_{k} converges then \sum_{k = 1}^{\infty} a_{k} converges . If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty} b_{k} diverges then \sum_{k = 1}^{\infty} a_{k} diverges .$

Step 3. Result

$Hence \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L Where L is any positive real number then either both converges or diverges .$

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If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit comparison test

Step 3. Result

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If limk→∞akbkWhereLis a positive finite number, what may we conclude about the two series?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit comparison test

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?