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

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)