Use the standard power series for sin(z) and cos(z) functions to determine the power series for sin(z) with centre 7/4.
Use the standard power series for sin(z) and cos(z) functions to determine the power series for sin(z) with centre 7/4.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9cc5b4ee-d88c-49e1-90ab-9eeeb95e2754%2F7aa0a4f2-978f-4bfe-9b4a-07511f9e49b9%2Fqgy3fr_processed.png&w=3840&q=75)
Transcribed Image Text:**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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