The coefficients of a power series for natural log

[the7joker7]

Homework Statement

The function f(x) =ln(10 - x) is represented as a power series in the form

f(x) = (sum from 0 to infinity) of c$$_{n}$$x$$^{n}$$

Find the first few coefficients in the power series.

The Attempt at a Solution

I know how to find the coefficients in a normal looking taylor series (for example, 3/(1 - 2x)^2 or something) but I don't have any idea where to start for a natural log...

for the record

C0 = 2.30258509299

C1 = -0.1

C2 = -0.005

C3 = -0.000333333333333

C4 = -2.5E-05

Radius of convergence = 10

Help?

 

[foxjwill]

There's a really nice trick for finding power series for functions of the form $$f(x) = ln(a+x)$$.

When you take the derivative of f, you get
$$f'(x)=\frac{1}{a+x}.$$
Since f'(x) expands to a geometric series, all you need to do find that and then take it's integral from 0 to x.

 

[Dick]

Homework Statement

The function f(x) =ln(10 - x) is represented as a power series in the form

f(x) = (sum from 0 to infinity) of c$$_{n}$$x$$^{n}$$

Find the first few coefficients in the power series.

The Attempt at a Solution

I know how to find the coefficients in a normal looking taylor series (for example, 3/(1 - 2x)^2 or something) but I don't have any idea where to start for a natural log...

for the record

C0 = 2.30258509299

C1 = -0.1

C2 = -0.005

C3 = -0.000333333333333

C4 = -2.5E-05

Radius of convergence = 10

Help?

You are only asked to find the first few coefficients in the power series. I think you are doing quite well. What's the problem?

 

[the7joker7]

No, I was able to find the answer afterwards by getting the question wrong. I don't know how to derive the answer from the question.

Well, I do now though. Thank you!