Describe the pattern in each sequence 21, 16, 11, 6, 1, -4
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Taylor series is
To find the Taylor series for f(x) = ln(x) centering at 9, 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
f(x) = ln(x)
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Since we need to have it centered at 9, we must take the value of f(9), and so on.
f(9) = ln(9)
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Following the pattern, we can see that for ,
This applies for n ≥ 1, Expressing f(x) in summation, we have
Combining ln2 with the rest of series, we have
Taylor series is
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