Q 54. Explain why the series is not a ... [FREE SOLUTION]

Chapter 8: Q 54. (page 670)

Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$

Short Answer

Expert verified

The value of $x$ for which the seriesrole="math" localid="1649519156951" ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$ converges when $x 鈭� R$ .

Step by step solution

Step 1. Given information.

The given power series is ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$ .

Step 2. Find the values of x for which the given series converge.

Since, the power of the $x$ is negative. So the series is not power series.

Now, the value of $b_{k}$ is

$b_{k} = {(- 1)}^{k} \frac{1}{k!} x^{- k}$ and $b_{k + 1} = {(- 1)}^{k + 1} \frac{1}{(k + 1)!} x^{- (k + 1)}$

Thus,

$\lim_{k 鈫� 鈭�} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k 鈫� 鈭�} |\frac{{(- 1)}^{k + 1} \frac{1}{(k + 1)!} x^{- (k + 1)}}{{(- 1)}^{k} \frac{1}{k!} x^{- k}}| = \lim_{k 鈫� 鈭�} |\frac{- 1}{x (k + 1)}|$

So, for $k 鈫� 鈭�$ the value of limit will always be zero.

Therefore the value of $x$ for which the series ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$ converges when $x 鈭� R$ .

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91影视

Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the values of x for which the given series converge.

One App. One Place for Learning.

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91影视

Explain why the series is not a power series in x-x0.Then use the ratio test for absolute convergence to find the values of xfor which the given series converge 鈭�k=0鈭�-1k1k!x-k

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the values of x for which the given series converge.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Statistics

Logic and Functions

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge ${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{1}{k!} x^{- k}$