# How do you find the nth Taylor polynomials for f(x) = ln x centered about a=1?

We have:

and in general:

So that:

The Taylor series is then:

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To find the nth Taylor polynomial for ( f(x) = \ln x ) centered about ( a = 1 ), you would first need to find the nth derivative of ( f(x) ) and evaluate each derivative at ( x = 1 ). The nth Taylor polynomial is then given by the formula:

[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n ]

For ( f(x) = \ln x ), the nth derivative is:

[ f^{(n)}(x) = (-1)^{n-1} (n-1)! \cdot \frac{1}{x^n} ]

Evaluating at ( x = 1 ), you get:

[ f^{(n)}(1) = (-1)^{n-1} (n-1)! ]

Thus, the nth Taylor polynomial for ( f(x) = \ln x ) centered about ( a = 1 ) is:

[ P_n(x) = \ln 1 + \frac{1}{1}(x-1) - \frac{1}{2 \cdot 1}(x-1)^2 + \frac{1}{3 \cdot 1}(x-1)^3 - \cdots + (-1)^{n-1} \frac{(x-1)^n}{n} ]

Simplified to:

[ P_n(x) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - \cdots + (-1)^{n-1} \frac{(x-1)^n}{n} ]

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To find the nth Taylor polynomial for ( f(x) = \ln x ) centered about ( a = 1 ), you can use the Taylor series expansion formula:

[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots ]

For ( f(x) = \ln x ), the derivatives are:

[ f(x) = \ln x ] [ f'(x) = \frac{1}{x} ] [ f''(x) = -\frac{1}{x^2} ] [ f'''(x) = \frac{2}{x^3} ] [ f^{(4)}(x) = -\frac{6}{x^4} ] [ \cdots ]

Evaluate these derivatives at ( x = a = 1 ) to find the coefficients for the Taylor polynomial. Since ( f(1) = \ln 1 = 0 ), the Taylor polynomial simplifies:

[ P_n(x) = 0 + \frac{1}{1}(x-1) - \frac{1}{2}(x-1)^2 + \frac{2}{3}(x-1)^3 - \frac{6}{4!}(x-1)^4 + \cdots ]

You can continue this pattern to find the nth Taylor polynomial for ( f(x) = \ln x ) centered about ( a = 1 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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