The coefficients of a power series for natural log

In summary, the conversation is about finding the first few coefficients in the power series of the function f(x) = ln(10-x). The attempt at a solution involved using a trick for finding power series for functions of the form f(x) = ln(a+x). The final output showed the first few coefficients found by using this method.
  • #1
the7joker7
113
0

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[tex]_{n}[/tex]x[tex]^{n}[/tex]

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?
 
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  • #2
There's a really nice trick for finding power series for functions of the form [tex]f(x) = ln(a+x)[/tex].

When you take the derivative of f, you get
[tex]f'(x)=\frac{1}{a+x}.[/tex]
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.
 
  • #3
the7joker7 said:

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[tex]_{n}[/tex]x[tex]^{n}[/tex]

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?
 
  • #4
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!
 

Related to The coefficients of a power series for natural log

What is the formula for finding the coefficients of a power series for natural log?

The formula for finding the coefficients of a power series for natural log is cn = (-1)n+1 / n.

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

The radius of convergence for a power series of natural log can be determined by using the ratio test. This involves taking the limit as n approaches infinity of the absolute value of cn+1 / cn. If the limit is less than 1, the series will converge; if the limit is greater than 1, the series will diverge; and if the limit is equal to 1, further testing is needed.

Can the coefficients of a power series for natural log be used to approximate the value of ln(x)?

Yes, the coefficients of a power series for natural log can be used to approximate the value of ln(x). The more terms that are included in the series, the more accurate the approximation will be.

What is the significance of the coefficients of a power series for natural log?

The coefficients of a power series for natural log represent the expansion of the natural log function into an infinite sum of powers of x. They allow for the evaluation of the natural log function at any value of x, even if the value is not a rational number.

Are there any patterns or relationships between the coefficients of a power series for natural log?

Yes, there are patterns and relationships between the coefficients of a power series for natural log. For example, the coefficients alternate in sign and decrease in magnitude as n increases. Also, the sum of the coefficients from n=1 to infinity is equal to ln(2).

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