Is the Maclaurin series for sin(2x) the same as the one for sin(x)?

In summary, a Maclaurin series is a type of power series expansion used to approximate a mathematical function by using powers of x as variables. To find the Maclaurin series, the derivatives of the function are evaluated at x=0 and substituted into a general formula. The purpose of using a Maclaurin series is to simplify a complicated function for easier evaluation. However, a Maclaurin series can only accurately represent a function if it is infinitely differentiable at x=0. The accuracy of a Maclaurin series approximation depends on the number of terms used, and can be determined by comparing it to the actual function at a given point.
  • #1
tandoorichicken
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We learned that the Maclaurin series for sin(x) was

[tex]\sum^{\infty}_{n=0} (-1)^n \frac{x^{2n+1}}{(2n+1)!} [/tex]

Is the Maclaurin series for sin(2x) the same, except with x replaced by 2x?
 
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  • #3


Yes, the Maclaurin series for sin(2x) is the same as the one for sin(x), except with x replaced by 2x. This is because the Maclaurin series for sin(x) is derived from the Taylor series for sin(x), which is a generalization that can be applied to any function. So, when we substitute 2x for x in the Maclaurin series for sin(x), we are essentially just plugging in a different value for x and the series remains the same. This is a useful property of Maclaurin series, as it allows us to easily find the series for functions that are related to each other by a simple substitution.
 

1. What is a Maclaurin series?

A Maclaurin series is a type of power series expansion that approximates a mathematical function using powers of x as the variables. It is named after Scottish mathematician Colin Maclaurin.

2. How do you find the Maclaurin series of a function?

To find the Maclaurin series of a function, you must first take the derivatives of the function and evaluate them at x=0. Then, substitute those values into the general formula for a Maclaurin series, which is given by f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ...

3. What is the purpose of using a Maclaurin series?

The purpose of using a Maclaurin series is to approximate a complicated function with a simpler one that can be easily evaluated. This is especially useful for calculating values of a function at points that are difficult to compute directly.

4. Can a Maclaurin series represent any function?

No, a Maclaurin series can only represent a function if that function is infinitely differentiable at x=0. If a function has a discontinuity or a singularity at x=0, then its Maclaurin series will not accurately represent the function.

5. How do you determine the accuracy of a Maclaurin series approximation?

The accuracy of a Maclaurin series approximation depends on the number of terms used in the series. The more terms that are included, the closer the approximation will be to the actual function. To determine the accuracy, you can compare the value of the approximation to the value of the actual function at a given point.

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