Chapter 8: Q. 65 (page 671)

Prove that if $(x_{0} - ρ, x_{0} + ρ]$ is the interval of convergence for the series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ , then the series converges conditionally at $x_{0} + p$ .

Short Answer

Expert verified

Ans: Therefore, the series converges conditionally at $x_{0} + p$

Step by step solution

Step 1. Given formation.

given,

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

Step 2. Consider the power series ∑k=0∞ akx−x0k and x0−ρ,x0+ρ is the interval of convergence of the series.

At $x_{0} + p$ , the series would become

$\sum_{k = 0}^{\infty} |a_{k} {((x_{0} + ρ) - x_{0})}^{k}| = \sum_{k = 0}^{\infty} |a_{k} ρ^{k}|$

This is precisely the series of absolute values when the original series is evaluated at $x_{0} + p$

Therefore, the series converges conditionally at $x_{0} + p$ .

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Prove that if $(x_{0} - ρ, x_{0} + ρ]$ is the interval of convergence for the series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ , then the series converges conditionally at $x_{0} + p$ .

Short Answer

Step by step solution

Step 1. Given formation.

Step 2. Consider the power series ∑k=0∞ akx−x0k and x0−ρ,x0+ρ is the interval of convergence of the series.

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Prove that if x0−ρ,x0+ρis the interval of convergence for the series ∑k=0∞ akx−x0k, then the series converges conditionally at x0+p.

Short Answer

Step by step solution

Step 1. Given formation.

Step 2. Consider the power series ∑k=0∞ akx−x0k and x0−ρ,x0+ρ is the interval of convergence of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove that if $(x_{0} - ρ, x_{0} + ρ]$ is the interval of convergence for the series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ , then the series converges conditionally at $x_{0} + p$ .