# How do you find the power series representation for the function #f(x)=ln(5-x)# ?

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To find the power series representation for the function f(x) = ln(5 - x), you can use the Taylor series expansion. The Taylor series expansion for ln(1 + x) is well-known:

ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ...

However, to use this expansion for ln(5 - x), we need to rewrite it in the form ln(1 + x). We can do this by noticing that 5 - x can be written as 1 - (-x/5). So:

ln(5 - x) = ln(1 - (-x/5)).

Now, we can use the Taylor series expansion for ln(1 + x):

ln(1 - (-x/5)) = -x/5 - (-x/5)^2/2 - (-x/5)^3/3 - (-x/5)^4/4 + ...

This can be simplified to:

= -x/5 - x^2/50 - x^3/750 - x^4/18750 + ...

Thus, the power series representation for f(x) = ln(5 - x) is:

ln(5 - x) = -x/5 - x^2/50 - x^3/750 - x^4/18750 + ...

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