Power series representation of 10xarctan(5x).

In summary, the function f(x)=10xarctan(5x) can be represented as a power series with coefficients given by summation from n=0 to infinity of (-1)^n (10x)(5x)^2n+1/(2n+1). This can be simplified to 50x^2 + 10x(5x)^3/3-10x(5x)^5/5+... in order to find the first few (5) coefficients in the power series. Despite some confusion from others, it is possible to multiply the entire expression by 10x.
  • #1
Boruedu
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Homework Statement



The function f(x)=10xarctan(5x) is represented as a power series
http://img464.imageshack.us/img464/4131/formub5.jpg
Find the first few(5) coefficients in the power series.

Homework Equations



I already know that the representation of arctanx is summation from n=0 to infinity of (-1)^n x^(2n+1)/(2n+1).

The Attempt at a Solution



So I plugged in 5x and then multiplied the entire thing by 10x.

I ended up with summation n=0 to infinity (-1)^n (10x)(5x)^2n+1/(2n+1) = 50x^2 + 10x(5x)^3/3-10x(5x)^5/5+...

I've asked a few people, and while they don't seem to remember power series very well, the only issue they could point out is that you can't multiply the whole thing by 10x. Is that true? If it is... then how do I go about solving this?
 
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  • #2
Perhaps those few people remember even less than they thought. Because you certainly can do that.
 

1. What is a power series representation?

A power series representation is a mathematical expression that represents a function as an infinite sum of powers of a variable. It is often used to approximate functions and calculate values that are difficult to obtain directly.

2. How is 10xarctan(5x) represented as a power series?

The power series representation of 10xarctan(5x) is: 10xarctan(5x) = x + (25/3)x^3 + (125/5)x^5 + (625/7)x^7 + ...

3. What is the radius of convergence for this power series?

The radius of convergence for this power series is 1/5, which means that the series will converge for all values of x within 1/5 units from the center, x=0. This can be determined by using the ratio test.

4. How accurate is the power series representation of 10xarctan(5x)?

The accuracy of the power series representation depends on the number of terms used. The more terms that are included, the more accurate the approximation will be. However, there will always be a margin of error as the series is an infinite sum and cannot be calculated exactly.

5. What are some applications of this power series representation?

This power series representation can be used in various mathematical calculations, such as finding the value of 10xarctan(5x) for a specific value of x, or approximating the behavior of other functions that are similar to 10xarctan(5x). It can also be used in physics and engineering problems that involve trigonometric functions.

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