Find the taylor series for f(x centered at the given value of
a. [assume that f has a power series expansion. do not show that rn(x ? 0. ] f(x = ln x, a = 5

To find the Taylor series for f(x) = ln(x) centering at 5, we need to observe the pattern for the first four derivatives of f(x). From there, we can create a general equation for f(n). Starting with f(x), we have


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Since we need to have it centered at 5, we must take the value of f(5), and so on. 


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Following the pattern, we can see that for ,



This applies for . Expressing f(x) in summation, we have 



Combining ln2 with the rest of series, we have