Yes it looks correct. And you are right about the reason you don't differentiate those terms, they are just constants.
Hi,
I am not too good with this stuff yet, I am still learning, so I have some questions about why I am doing the things I am.
I am supposed to find a power series representation of f(x) = 1/(4+x)^2 and find its radius of convergence
So I realize that the integral of f(x) = -1/(4+x) = (-1/4)*[1/(1+(x/4))] = (-1/4)*[1/(1-(-x/4))]. This is a geometric series with a = 1 and r = -x/4, which means that it has a radius of convergence = |x|/4 < 1 => radius of convergence = 4
(-1/4)*[1/(1-(-x/4)) can be rewritten as (-1/4)[SERIES(n=0,inf) (-x/4)^n]
and when I distribute and multiple I get
SERIES(n=0,inf) [-1*(-1)^n(x^n)]/4^(n+1)
So since I found the series of the integral of f(x) I have to take its derivative, right?
The derivative of the above series is
SERIES(n=1,inf) [-1*(-1)^n(n*x^(n-1))]/4^n+1
Why didn't I have to do the power rule and differentiate the (-1)^n and 4^(n+1), is it because they don't vary by x?
Is it because I want the "a," the first term in the geometric series to = 1? And it would equal 0 if n was 0.
Why did I have to change the bounds of the series to n=1 instead of starting at n=0?
So to get back to having a bound starting at n=0 I add +1 to all the n values.
SERIES(n=0,inf) [-1*(-1)^(n+1)((n+1)*x^n)]/4^n+2
This is equal to SERIES(n=0,inf) [(-1)^n*(n+1)*x^n)]/4^n+2
And that's what I am told is the right answer. So could someone just make sure I am doing the steps right and help with the questions I had? I really appreciate it, bear with me, I am pretty confused by this right now, because my professor hasn't really gone over this and I am having trouble finding resources :s.
Thank you I really appreciate any help.