Q. 12 Use Exercise 11 to explain why t... [FREE SOLUTION]

Chapter 7: Q. 12 (page 639)

Use Exercise 11 to explain why the root test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k} in which a_{k}$ is a power function .

Short Answer

Expert verified

Hence proved that the root test will be inconclusive because the value of $L = 1$ and root test is conclusive at $L = 1$ .

Step by step solution

Step 1. Given Information.

We are given ${鈭�}_{k = 1}^{鈭�} a_{k} .$

Step 2. Root test.

Now, using Root test,

$\lim_{k 鈫� 鈭�} {(A x^{r})}^{\frac{1}{k}} = \lim_{k 鈫� 鈭�} (A)^{\frac{1}{x}} \lim_{k 鈫� 鈭�} {(x^{r})}^{\frac{1}{k}} = \lim_{k 鈫� 鈭�} (A)^{\frac{1}{k}} \lim_{k 鈫� 鈭�} (x^{\frac{r}{k}}) Since, a^{x} = e^{\ln (a^{x})}, therefore, x^{\frac{r}{k}} = e^{\ln (x^{\frac{r}{2}})} = e^{\frac{r}{k} \ln (x)} N o w, \lim_{k 鈫� 鈭�} (x^{\frac{r}{k}}) = \lim_{k 鈫� 鈭�} e^{\frac{r}{k} \ln (x)} = e^{\lim_{i 鈫� 鈭�} (\frac{r}{k} \ln (x))} = 1$

Therefore,

$\lim_{k 鈫� 鈭�} {(A x^{r})}^{\frac{1}{k}} = 1 \lim_{k 鈫� 鈭�} a_{k} {}^{\frac{1}{k}}= 1 L = 1 T h e r e f o r e, i t i s a i n c o n c l u s i v e .$

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91影视

Use Exercise 11 to explain why the root test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k} in which a_{k}$ is a power function .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Root test.

One App. One Place for Learning.

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91影视

Use Exercise 11 to explain why the root test will be inconclusive for every series 鈭�k=1鈭�akin whichakis a power function .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Root test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Pure Maths

Mechanics Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Use Exercise 11 to explain why the root test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k} in which a_{k}$ is a power function .