Q. 12 Show that the power series 鈭慿=... [FREE SOLUTION]

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Chapter 8: Q. 12 (page 669)

Show that the power series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$ converges absolutely when $x = 1$ and when $x = - 1$ . What does this behavior tell you about the interval of convergence for the series?

Short Answer

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Ans: The power series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$ has the interval of convergence $[- 1, 1]$

Step by step solution

Step 1. Given information.

given,

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$

Step 2. Evaluate the series when x=1

So,

$\begin{aligned} {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k} = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} (1)^{k} \\ = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} \end{aligned}$

So, for $x = 1$ , we have the alternating harmonic series which converges conditionally.

Step 3. We evaluate the series when x=-1

So,

$\begin{aligned} {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k} = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} (鈭� 1)^{k} \\ = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{2 k}}{k^{2}} \\ = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{k^{2}} \end{aligned}$

So, for $x = - 1$ , we have the alternating harmonic series which converges conditionally.

Step 4. Thus,

Therefore, the power series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$ has the interval of convergence $[- 1, 1]$ .

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91影视

Show that the power series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$ converges absolutely when $x = 1$ and when $x = - 1$ . What does this behavior tell you about the interval of convergence for the series?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate the series when x=1

Step 3. We evaluate the series when x=-1

Step 4. Thus,

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91影视

Show that the power series 鈭�k=1鈭�鈥�(鈭�1)kk2xkconverges absolutely when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate the series when x=1

Step 3. We evaluate the series when x=-1

Step 4. Thus,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

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Calculus

Study anywhere. Anytime. Across all devices.

Show that the power series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{k^{2}} x^{k}$ converges absolutely when $x = 1$ and when $x = - 1$ . What does this behavior tell you about the interval of convergence for the series?