Power Series Help: Find Coefficients & Radius of Convergence for 4x/(7+x)

In summary: So C0=0 and C1=4/7.In summary, the given function f(x)=4x/(7+x) can be written as a power series with coefficients of 0 and 4/7, with a radius of convergence of 7. The coefficients are found by expanding 1/(7+x) in a power series and then multiplying by 4x. The coefficient of the 0 power is always 0, while the coefficient of the 1 power is 4/7.
  • #1
snoggerT
186
0
f(x)=4x/(7+x). Find the first few coefficients and radius of convergence



sum (n=0 to infinity) CnX^n



The Attempt at a Solution


I set up the equation into the form of a power series and got:

(-1)^n*(4)^n*(x/7)^(n+1)

But that doesn't seem to be right because I can't get the coefficients right. I know the first one is 0, but what I have doesn't give me that. the RoC is 7. What am I doing wrong?
 
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  • #2
If you do it correctly in terms of a power series it's not 4^n. It's just 4. Just expand 1/(7+x) in power series and then multiply by 4x. Right, there is no zero term.
 
  • #3
Dick said:
If you do it correctly in terms of a power series it's not 4^n. It's just 4. Just expand 1/(7+x) in power series and then multiply by 4x. Right, there is no zero term.

- Thanks. That fixes the problem. Though webworks is pretty dumb as always. It list C0=0 and C1=4/7. That doesn't make sense to me since plugging in 0 gives you 4/7.
 
  • #4
snoggerT said:
- Thanks. That fixes the problem. Though webworks is pretty dumb as always. It list C0=0 and C1=4/7. That doesn't make sense to me since plugging in 0 gives you 4/7.

It's correct. 4x/(7+x)=4x*(1/7-x/7^2+x^2/7^3...). If C0 is the power of x^0, it's zero. If C1 is power of x^1, then it's 4/7. Etc.
 
  • #5
Dick said:
It's correct. 4x/(7+x)=4x*(1/7-x/7^2+x^2/7^3...). If C0 is the power of x^0, it's zero. If C1 is power of x^1, then it's 4/7. Etc.

- I see. I need to go back and review it more. I thought it was assumed that anything raised to the 0 power is 1.
 
  • #6
You are right! The coefficient of the 0 power is the coefficient of 1=x^0. It's 0. 4/7 is the coefficient of x=x^1.
 
Last edited:

What is a power series?

A power series is an infinite series of the form ∑n=0 an(x-c)n, where an are coefficients, x is the variable, and c is a constant. It is used to represent a function as a sum of infinitely many terms.

How do you find the coefficients of a power series?

To find the coefficients of a power series, you can use the formula an = f(n)(c)/n!, where f(x) is the function represented by the power series and c is the center of the series. This formula involves taking derivatives of the function at the center point and dividing by the factorial of n.

What is the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center point to the nearest point where the series converges. It is usually denoted by R and can be calculated using the formula R = 1/L, where L is the limit of the ratio of consecutive coefficients.

How do you determine the radius of convergence of a power series?

To determine the radius of convergence, you can use the ratio test, which involves taking the limit of the ratio of consecutive coefficients as n approaches infinity. If the limit is less than 1, the series converges. If it is greater than 1, the series diverges. If it is equal to 1, further testing is needed.

How do you apply these concepts to find the coefficients and radius of convergence for 4x/(7+x)?

To find the coefficients of this power series, we can use the formula an = f(n)(c)/n!, where c = 0 and f(x) = 4x/(7+x). This results in an = (-1)n4n+1/7n+1. To find the radius of convergence, we can use the ratio test, which gives us R = 7. Therefore, the power series converges for all values of x within a distance of 7 from the center point x = 0.

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