# What is the power series of ##f(x)= ln(5-x)##? What is it’s radius of convergence?

Since the ##int1/(5-x)##= ##ln(5-x)## get the power series representation of ##1/(5-x)##.
Algebraically manipulate the equation to get the equation for the sum of the (which is ##Sn= a/(1-r)## ).
Using this we could use the power series representation of the geometric series ##sumar^(n)## such that ##a=1/5## and ##r=x/5##
Substituting this in the pwer series equation, we get
##sum(1/5)(x/5)^n##.
Integrate this power series representation to get thepower series of ##ln(5-x)##.
##int####sum(1/5)(x/5)^n##. = ##sum(1/(5(n+1)))(x/5)^(n+1)##.
Use Ratio Test on ##sum(1/(5(n+1)))(x/5)^(n+1)## to get the Radius and Interval of Convergence.
For IOC, do not foget to determine convergence/divergence at end points to determine if included in the ##domf## or not
In getting Radius of Convergence, it must be in this form,
##abs(x-c)####< 1## such that the series is centered at c. R=5 (radius of convergence)

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