Chapter 8: Q. 5 (page 669)

Explain why $\sum_{k = 0}^{\infty} {(x - k)}^{k}$ is not a power series.

Short Answer

Expert verified

The base of the factors ${(x - k)}^{k}$ varies with the index.

Step by step solution

Step 1. Given information.

Consider the given question,

$f (x) = \sum_{k = 0}^{\infty} {(x - k)}^{k}$

Step 2. Explain the series.

If $\sum_{k = 0}^{\infty} a_{k}$ is a sequence of real numbers and x be a variable and $x_{0}$ be a real number. Then a power series in $x - x_{0}$ is a series of the form localid="1649342585929" ${\sum a_{k}}_{k = 0}^{\infty} {(x - x_{0})}^{k}$

Rewriting the series,

$f {\sum a_{k}}_{k = 0}^{\infty} {(x - x_{0})}^{k} = a_{0} + a_{1} (x - x_{0}) + a_{2} {(x - x_{0})}^{2} + a_{3} {(x - x_{0})}^{3} + . . .$

The series of $f (x)$ is not a power series, this is because $x_{0}$ in the series $f (x) = \sum_{k = 0}^{\infty} {(x - k)}^{k}$ , that is k, is not a constant real number.

We can say that the base of the factors ${(x - k)}^{k}$ varies with the index.

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Explain why $\sum_{k = 0}^{\infty} {(x - k)}^{k}$ is not a power series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explain the series.

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Explain why ∑k=0∞x-kk is not a power series.

Short Answer

Step by step solution

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Step 2. Explain the series.

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Most popular questions from this chapter

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Explain why $\sum_{k = 0}^{\infty} {(x - k)}^{k}$ is not a power series.