Maclaurin Series for Expanding sin(2x)^2: Step-by-Step Guide

• vabamyyr
In summary, the conversation is about expanding f(x)= (sin2x)^2 into Maclaurin series and finding a suitable identity to use. The identity \sin^2 x=\frac{1}{2}(1-\cos 2x)\sin^2 2x=\frac{1}{2}(1-\cos 4x)=0.5-0.5cos4x was found to be useful in this case and the coefficient 0.5 was determined to be cancelled out in the final expansion.
vabamyyr
i have trouble expandind f(x)= (sin2x)^2 into Maclaurin series
for sin(x) Maclaurin series is

$$\sum^{\infty}_{n=0} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$

probably the key is to change (sin2x)^2 into new shape. I found that

(sin2x)^2=2sin(2x^2), but that coefficent 2 is bothering me, what to do?

vabamyyr said:
(sin2x)^2=2sin(2x^2), but that coefficent 2 is bothering me, what to do?
That relation does not hold (the left side is always positive, the right side isn't).

But you can use another identity:

$$\sin^2 x=\frac{1}{2}(1-\cos 2x)$$

$$\sin^2 2x=\frac{1}{2}(1-\cos 4x)=0.5-0.5cos4x$$

and now if i apply for cosx maclaurin series expansion considering the function f(x)= -0,5cos4x i get the right answer but I am puzzled, where does the coefficent 0,5 go? i don't have to count that??

Last edited:
The constant term of the series for sin2(2x) is 0.
The 0.5 cancels the -0.5 from the expansion of -0.5*cos(4x).

Last edited:

What is a Maclaurin series?

A Maclaurin series is a type of mathematical series that approximates a function by using a polynomial. It is named after Scottish mathematician Colin Maclaurin and is a special case of a Taylor series.

Why are Maclaurin series important?

Maclaurin series are important because they allow us to approximate any function with a polynomial, making it easier to perform calculations and make predictions in mathematics and science. They are also used in various fields such as physics, engineering, and economics.

How do I find the Maclaurin series of a function?

To find the Maclaurin series of a function, you can use the Taylor series formula and substitute in x = 0. This will give you the specific formula for the Maclaurin series of that function. Alternatively, you can also use known Maclaurin series for common functions such as sine, cosine, and exponential functions.

What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is a special case of a Taylor series, where the center of the series is at x = 0. This means that the terms of a Maclaurin series are in the form of x^n, while a Taylor series can have a center at any point, resulting in terms of (x-a)^n.

How accurate is a Maclaurin series approximation?

The accuracy of a Maclaurin series approximation depends on the number of terms used in the series. The more terms included, the closer the approximation will be to the actual function. However, since it is an approximation, there will always be some degree of error. The error can be estimated using the remainder term of the series.

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