Solving Taylor's Theorem for f(x)=ln(x+1) at x=1.5

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In summary, the conversation is about computing the minimum number of terms for a Taylor polynomial to approximate f(1.5) within .0001 using Taylor's theorem. There is some difficulty in finding the correct value for the (n+1)th derivative of ln(x + 1), and the suggestion to use the absolute value of the error and the first term omitted is mentioned. However, the person is still struggling to find the term at which the error is less than .00001, even after multiplying by 1/n!. They believe it should be 9 terms, but it is only reached after 10 terms. Assistance is requested.
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I'm computing the minimum number of terms for a Taylor polynomial to approximate f(1.5) within .0001 where f(x) = ln(x + 1) using Taylor's theorem, but I'm having a little trouble getting there. I keep coming up with the absolute value of the (n+1)th derivative of ln(x + 1) as (n!)/[(x+1)^(n+1)] in which case the largest value for any derivative of ln(x + 1) from 0 to x would be n! but if I use this with Taylor's Theorem I get (n!)[(1.5)^(n+1)] / (n+1)! < .0001 but this is not true for any n. Any help would be appreciated.
 
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Did you forget the 1/n! in Taylor's formula?
 
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Since the series is a alternating, why not use "the absolute value of the error is less than the absolute value of the first term omitted"?
 
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Well that's the problem benorin, I'm not looking for just the error, I already know what it should be. I'm looking for the term at which the error is less than .00001. In the case of the other post, even if I multiply by 1/n! I'm simply left with 1.5^(n+1) / (n+1)! which is less than .00001 only after 10 terms when it should be 9.
 

FAQ: Solving Taylor's Theorem for f(x)=ln(x+1) at x=1.5

What is Taylor's Theorem?

Taylor's Theorem is a mathematical theorem that allows us to approximate a function using a polynomial. It is based on the idea that any smooth function can be represented as an infinite sum of polynomial terms.

How is Taylor's Theorem used to approximate a function?

In order to use Taylor's Theorem, we must have a known value for the function and its derivatives at a specific point (usually denoted as "a"). Then, we can use the Taylor series expansion to find an approximation for the function at a different point (usually denoted as "x").

What is the Taylor series for f(x)=ln(x+1) at x=1.5?

The Taylor series for f(x)=ln(x+1) at x=1.5 is: ln(1.5+1) + (x-1.5)/1 - (x-1.5)^2/2 + (x-1.5)^3/3 - (x-1.5)^4/4 + ...

How do we solve for the coefficients in Taylor's Theorem?

To find the coefficients in Taylor's Theorem, we can use the formula: f^(n)(a)/n!, where f^(n) represents the nth derivative of the function evaluated at the point "a". We can then substitute these coefficients into the Taylor series expansion to approximate the function at a different point.

Why is it important to use Taylor's Theorem for approximation?

Taylor's Theorem allows us to approximate a function with a polynomial, which can be easier to work with and calculate compared to the original function. It also allows us to find more accurate approximations by including higher order terms in the Taylor series expansion.

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