Find the first five non-zero terms of power series representation centered at x=0 for the function below.

f(x)=ln(6-x)

The solution to the problem is as follows:

Start with the geometric series 

1/(1 - t) = Σ(n = 0 to ∞) t^n, convergent for |t| < 1. 

Integrate both sides from 0 to t: 

-ln(1 - t) = Σ(n = 0 to ∞) t^(n+1)/(n+1), convergent (save endpoint(s)) for |t| < 1. 

Let t = x/6: 

-ln(1 - x/6) = Σ(n = 0 to ∞) (x/6)^(n+1)/(n+1), convergent (save endpoint(s)) for |x/6| < 1. 

==> ln(1 - x/6) = -Σ(n = 0 to ∞) x^(n+1)/[6^(n+1) * (n+1)], convergent (save endpoint(s)) for |x| < 6. 

However, ln(1 - x/6) = ln((1/6)(6 - x)) = ln(1/6) + ln(6 - x) = ln(6 - x) - ln 6. 

Hence, we have 

ln(6 - x) = ln 6 - Σ(n = 0 to ∞) x^(n+1)/[6^(n+1) * (n+1)], which has R = 6.

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