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What is the radius of convergence? Would I be using ratio test?

What is the radius of convergence? Would I be using ratio test?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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What is the radius of convergence? Would I be using ratio test?
The image contains a mathematical expression involving an infinite series. The expression is as follows: \[ a \sum_{k=0}^{\infty} \frac{(-1)^k \cdot x^{2k+1}}{(2k + 1)!} \] This series is a summation starting from \( k = 0 \) to infinity, where: - \( (-1)^k \) represents alternating signs for each term. - \( x^{2k+1} \) is the variable \( x \) raised to the power of \( 2k+1 \). - \( (2k+1)! \) is the factorial of \( 2k+1 \), which is the product of all positive integers up to \( 2k+1 \). - \( a \) is a constant factor multiplying the entire sum. This type of series is characteristic of a Taylor or Maclaurin series, which is used to represent functions as an infinite sum of terms.
Transcribed Image Text:The image contains a mathematical expression involving an infinite series. The expression is as follows: \[ a \sum_{k=0}^{\infty} \frac{(-1)^k \cdot x^{2k+1}}{(2k + 1)!} \] This series is a summation starting from \( k = 0 \) to infinity, where: - \( (-1)^k \) represents alternating signs for each term. - \( x^{2k+1} \) is the variable \( x \) raised to the power of \( 2k+1 \). - \( (2k+1)! \) is the factorial of \( 2k+1 \), which is the product of all positive integers up to \( 2k+1 \). - \( a \) is a constant factor multiplying the entire sum. This type of series is characteristic of a Taylor or Maclaurin series, which is used to represent functions as an infinite sum of terms.
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