Power Series?

Question

F(x)=ln(5-x)......can be represented by the power series f(x)= sum


from n=0 to n=INF of (CnX˚) ....find the first 5 coefficients and the radius of convergence.....ANY HELP i dont understand power series.......(˚ is sposed to be an "n" exponent)

Answer 1

I can give you the answer or give you real help. What is it that you don't understand? E-mail me. And please since I'm being nice enough to answer, be nice enough to chose a best answer.


essentially you just have to plug into the formula presented int he wiki entry below. If you know how to do derivatives, there is very little to "understand", unless you have to prove this result true.
http://en.wikipedia.org/wiki/Taylor_series

Basically, you simply need to plug in Taylor's formula, which is based on the derivatives of the numbers below.

You should know the derivate of ln(5-x) is -1/(5-x).
(By the chain rule theorem.) The derivate of ln(u) is
1/u. The derivative of 5-x is -1. So the derivative of
ln(5-x) is 1/(5.-x)

Now the derivative of -1/(u) is 2/(U^2). So the derivative of
-1/(5-x) is 2/(5-x)^2

Etc

All you have to do then, is find a value of ln(u) that you know.
Everybody knows that ln(1) = 0. So chose that.
5-x = 1 mean x =4.

So the first co-effcient is 0!!!

No, for the next, do the same thing using x=4 for the next term.
1/(5-4) = 1!!!

...

Etc.

Last note:
A^B means A raised to the exponent
B.

As for the radius of convergence, you know that ln(0) goes to infinity. Thus your formula goes to hell once you reach
x = 5!!! So your radiuus of convergence is 5-4 which is equal to 1!!!