Understanding the Maclaurin Series of cos(x) for Use in Functions: A Guide

In summary, the Maclaurin series of cos(x) can be derived using various methods and can be applied to obtain the Maclaurin series for h(x)=(cos(3x)-1)/x^2 by simplifying the known series and expressing it in standard form.
  • #1
Muzly
8
0
I've looked at a number of explanations for the Maclaurin series of cos(x) yet none have given an easily understood answer, i was wondering if anyone has a way of explaining it when it is used as only a part of a function

eg. use that Maclaurin series of cos(x) to obtain the Macalurin series for

h(x)=(cos(3x)-1)/x^2
 
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  • #2
The formula for Maclaurin series is
[tex]f(x)=\sum_{k=0}^\infty \frac{x^k}{k!}f^{(k)}(x)[/tex]
as can be easily derived several ways
-integration by parts
-formal operator manipulation
-matching derivatives at zero
-complex contour integration
This can be applied to cos(x) as
[tex]\cos(x)=\sum_{k=0}^\infty \frac{x^k}{k!}\cos^{(k)}(0)=\sum_{k=0}^\infty \frac{x^k}{k!}\cos(k\pi/2)=\sum_{k=0}^\infty \frac{x^{2k}}{(2k)!}(-1)^k[/tex]
where the index is changed in the final step to indicate odd terms are 0
Often a known Maclaurin series can be used to find a closely related one.
Just apply the known series
h(x)=(cos(3x)-1)/x^2
[tex]h(x)=\frac{1}{x^2}\left(\sum_{k=0}^\infty \frac{(3x)^{2k}}{(2k)!}(-1)^k-1\right)[/tex]
simplify that into standard form.
 
  • #3


The Maclaurin series of cos(x) is a mathematical tool that allows us to approximate the value of the cosine function at any point on the x-axis. It is a special case of the Taylor series, which is a way of representing a function as an infinite sum of terms. The Maclaurin series specifically is a Taylor series centered at x=0, meaning it is a way of representing a function as an infinite sum of terms centered at the origin.

When using the Maclaurin series of cos(x) to obtain the Maclaurin series for a more complex function, such as h(x)=(cos(3x)-1)/x^2, we can use the known Maclaurin series of cos(x) to simplify the process. First, we can rewrite the given function as h(x)=cos(3x)/x^2 - 1/x^2. Then, using the Maclaurin series of cos(x), we can write cos(3x) as an infinite sum of terms centered at x=0. This gives us h(x)= (1 - (3x)^2/2! + (3x)^4/4! - ...) /x^2 - 1/x^2.

Next, we can simplify the first term using basic algebra to get h(x)= (1 - 9x^2/2 + 81x^4/24 - ...) /x^2 - 1/x^2. Finally, we can combine the two fractions and simplify to get the Maclaurin series for h(x)= 1/x^2 - 9x^2/2 + 81x^4/24 - ... This series can then be used to approximate the value of h(x) at any point on the x-axis.

In summary, the Maclaurin series of cos(x) is a useful tool for approximating the value of the cosine function at any point on the x-axis. When used as a part of a more complex function, it can be used to simplify the process of finding the Maclaurin series for that function. Overall, understanding the Maclaurin series can greatly aid in solving mathematical problems and analyzing functions.
 

1. What is a Maclaurin series?

A Maclaurin series is a special type of power series expansion that represents a function as an infinite sum of terms. It is named after Scottish mathematician Colin Maclaurin, who first introduced the concept in the 18th century.

2. How is a Maclaurin series different from a Taylor series?

A Maclaurin series is a special case of a Taylor series, where the expansion is centered at x = 0. This means that all the derivatives of the function are evaluated at x = 0. In general, a Taylor series can be centered at any point, not just x = 0.

3. What is the formula for a Maclaurin series?

The formula for a Maclaurin series is given by f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ... + (f(n)(0)/n!)x^n + ...

4. How is a Maclaurin series used in calculus?

A Maclaurin series is used to approximate a function as a polynomial, which can make it easier to integrate or differentiate. It can also be used to evaluate functions at values that are difficult to compute, such as irrational numbers or complex numbers.

5. What are some common applications of Maclaurin series?

Maclaurin series are commonly used in physics, engineering, and other fields to model and analyze real-world phenomena. They are particularly useful in situations where a function can be approximated by a polynomial, such as in the study of oscillations, waves, and other periodic phenomena.

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