Taylor Series Homework: Find Series for f(x)=sin x at a=pi/2

In summary, the conversation discusses finding the Taylor series for f(x) = sin x centered at a = pi/2. The process involves taking derivatives of the function and plugging in the given value for x. The purpose of the Taylor series is to prove that it converges to sin x on the given interval. Useful resources for understanding and calculating Taylor series are Wikipedia and MathWorld.
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rcmango
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Homework Statement



Find the Taylor series for f(x) = sin x centered at a = pi / 2

Homework Equations





The Attempt at a Solution



Taylor series is a new series for me.

I believe the first step is to start taking the derivative of the Taylor series.

f(x) = sinx
f'(x) = cosx
f''(x) = -sinx
f'''(x) = -cosx
f(4)x= sinx
...
f(n)x = sin(x)^n

now do i start plugging in the a = pi/2 for x?
Okay, not sure what I'm trying to prove with a taylor series, and why to use it, what's next.

Also, I have to prove that the series converges to sinx on (-infinity, infinity)

Thanks.
 
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Related to Taylor Series Homework: Find Series for f(x)=sin x at a=pi/2

What is a Taylor series?

A Taylor series is a representation of a function as an infinite sum of terms, where each term is a derivative of the function evaluated at a specific point.

What is the formula for a Taylor series?

The formula for a Taylor series is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

How do you find the Taylor series for a function?

To find the Taylor series for a function, you need to evaluate the function and its derivatives at a specific point, a. Then, plug in the values into the formula for a Taylor series to get the series representation of the function.

What is the Taylor series for f(x)=sin x at a=pi/2?

The Taylor series for f(x)=sin x at a=pi/2 is:
sin x = 1 - (x-pi/2)^2/2! + (x-pi/2)^4/4! - (x-pi/2)^6/6! + ...

What is the purpose of finding a Taylor series?

The purpose of finding a Taylor series is to approximate a function with polynomials, which can make it easier to perform calculations and analyze the behavior of the function at a specific point. It can also be used to find the value of the function at points where it is difficult to evaluate directly.

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