Help with power series representation

[bobbarkernar]

Homework Statement

find a power series representation for the function and determine the radius of convergence.

f(t)= ln(2-t)

Homework Equations

The Attempt at a Solution

i first took the derivative of ln(2-t) which is 1/(t-2)

then i tried to write the integral 1/(t-2) in the form of a power series i got the sum for n=0 to infinity of (t/2)^n i don't know if this is write if someone can please help me

 

[courtrigrad]

If [itex] f(t) = \ln(2-t) [/tex] then write this as follows:[itex] -\ln(2-t) = \int \frac{1}{2-t} \; dt [/tex]

How would you transform [itex] \frac{1}{2-t} [/tex] into a more recognizable form: i.e. $\frac{1}{1-t}$?

 

[bobbarkernar]

i would factor out a 1/2 so i would have (1/2)*1/(1-(1/2t))

 

[courtrigrad]

ok then what would you do?

 

[bobbarkernar]

i would write that as a power series:
(1/2)*the sum for n=0 to infinity of (1/2t)^n

 

[courtrigrad]

correct you have:

$$\frac{1}{2}\sum_{n=0}^{\infty} (\frac{t}{2})^{n}$$

 

[bobbarkernar]

so is that the answer??

 

[courtrigrad]

yes that is the answer.

or write it like this:

$$\sum_{n=0}^{\infty} \left(\frac{t^{n}}{2^{n+1}}\right)$$

 

[bobbarkernar]

ok thank you very much

 

[joeG215]

correct you have:

$$\frac{1}{2}\sum_{n=0}^{\infty} (\frac{t}{2})^{n}$$

I'm confused... What happens to the integral?