**Title:** Determining the Power Series for sin(z) Centered at π/4**Content:**To find the power series for sin(z) with its center at π/4, we can use the standard power series expansions of the sine and cosine functions. The standard power series for sin(z) is:\[ \sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!} \]And for cos(z):\[ \cos(z) = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!} \]To determine the power series representation for sin(z) centered at z = π/4, we can substitute z with (z - π/4) in the series expansion of sin(z). This involves computing the derivatives of sin(z) evaluated at π/4 and constructing the series from these values.By using the derivatives and applying the Taylor series formula: \[ f(z) = \sum_{n=0}^{\infty} \frac{f^n(\pi/4)}{n!} (z - \pi/4)^n \]This results in a shifted power series expansion around the center point π/4. The process involves calculating values for each derivative of sin, evaluating them at π/4, and using these to formulate the series.This method is a powerful tool for approximating trigonometric functions within a desired interval, providing accurate results when considering the convergence of the series.

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Answered: Use the standard power series for…

Use the standard power series for sin(z) and cos(z) functions to determine the power series for sin(z) with centre 7/4.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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