Chapter 7: Q. 13 (page 639)

Explain how you could adapt the root test to analyze a series $\sum_{k = 1}^{\infty} a_{k}$ in which the terms of the series are all negative.

Short Answer

Expert verified

Hence proved.

Step by step solution

Step 1. Given information.

We are given $\sum_{k = 1}^{\infty} a_{k}$ .

Step 2. Explanation.

According to the root test,

$1. If L < 1 series converges. 2. If L > 1 series diverges. 3. If L = 1 the test is inconclusive.$

Where, Series is $\sum_{k = 1}^{\infty} a_{k}$ and $L = \lim_{k \to \infty} {(a_{k})}^{\frac{1}{k}}$ .

But if the series has all the terms negative, then negative sign can be adjusted and Root test can be used on the series $\sum_{k = 1}^{\infty} (- a_{k})$ .

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Explain how you could adapt the root test to analyze a series $\sum_{k = 1}^{\infty} a_{k}$ in which the terms of the series are all negative.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

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Explain how you could adapt the root test to analyze a series∑k=1∞akin which the terms of the series are all negative.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

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Explain how you could adapt the root test to analyze a series $\sum_{k = 1}^{\infty} a_{k}$ in which the terms of the series are all negative.