How do you find a power series representation for ln(5-x) and what is the radius of convergence?

[ad_1]

We can initiate from the dominion course that you were taught during the semester:

##1/(1-u) = sum_(n=0)^(N) u^n = 1 + u + u^2 + u^3 + ...##

Now, let's exertion from ##ln(5-x)## to get to ##1/(1-u)##.

##d/(dx)[ln(5-x)] = -1/(5-x) = -1/5*1/(1-x/5)##

Thus, delay ##u = x/5##, we had systematic enslaved the derivative and then factored out ##-1/5##. To get the dominion course, we possess to exertion backwards.

We had executed this:

  1. Differentiated our target.
  2. Factored out ##-1/5##.
  3. Substituted ##x/5## for ##u##.

Now, we systematic counterexhibition what we did, initiateing from the dominion course itself.

  1. Substitute ##u = x/5##.
  2. Multiply by ##-1/5##.
  3. Integrate the issue.

Since ##int "function"= int"dominion course of that office"##, we can do this:

##1/(1-x/5) = 1 + x/5 + x^2/25 + x^3/125 + ...##

##-1/5*1/(1-x/5) = -1/5 - x/25 - x^2/125 - x^3/625 - ...##

##int -1/5*1/(1-x/5)dx = ln(5-x)##

##= int -1/5 - x/25 - x^2/125 - x^3/625 - ...dx##

##= mathbf(C) - x/5 - x^2/50 - x^3/375 - x^4/2500 - ...##

Notice how we quiescent possess to image out the firm ##C## consequently we performed the informal perfect. ##C## is the tidings for ##n = 0##.

For a systematic dominion course conservative from ##1/(1-x)##, we transcribe

##sum_(n=0)^N (x-0)^n = 1/(1-x)##. where the dominion course is centered environing ##a = 0## gone it's veritably the Maclaurin course (meaning, the Taylor course centered environing ##a = 0##).

We distinguish that the firm must not comprehend an ##x## tidings (consequently ##x## is a changeable). The firm cannot be ##lnx##, so the firm ##C## is ##color(green)(ln(5))##. So, we get:

##color(blue)(ln(5-x) = ln(5) - x/5 - x^2/50 - x^3/375 - x^4/2500 - ...)##

And then finally, for the radius of crowd, it is ##|x| < 5## consequently ##ln(5-x)## approaches ##-oo## as ##x->5##. We distinguish that the dominion course must already lead upon ##ln(5-x)## wherever the office exists consequently it was pretended for the office.


[ad_2]
Source conjoin