Q.o. Read the section and make your o... [FREE SOLUTION]

Chapter 7: Q.o. (page 630)

Read the section and make your own summary of the material

Short Answer

Expert verified

1) Comparison test : Let ${鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭� b_{k}}_{k = 1}^{鈭�}$ be two series with nonnegative terms such that $0 鈮� a_{k} 鈮� b_{k}$ for every positive integer k. If the series ${鈭� b_{k}}_{k = 1}^{鈭�}$
converges , then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

2) The limit comparison test

Let ${鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms .

a) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where L is any positive real number, then either both series converges or both series diverges.

b) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

c) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges..

Step by step solution

Given,The information in the section of book

1) Comparison test : Let ${鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with nonnegative terms such that $0 鈮� a_{k} 鈮� b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges , then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

step 2

2) The limit comparison test

Let ${鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms .

a) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where L is any positive real number, then either both series converges or both series diverges.

b) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

c) If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges, then ${鈭�}_{k = 1}^{鈭�} a_{k} d i v e r g e s$ .

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Read the section and make your own summary of the material

Short Answer

Step by step solution

Given,The information in the section of book

step 2

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