Power series coefficients for 4x/(7+x) and radius of convergence

[snoggerT]

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?

 

[Dick]

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.

 

[snoggerT]

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.

 

[Dick]

- 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.

 

[snoggerT]

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.

 

[Dick]

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.