Approximating mac/taylor series to z decimal accuracy

• frozenguy
In summary, to approximate the value of tan^{-1}\left(\frac{1}{2}\right) to three decimal place accuracy using the Mac series, you need to use the third term of the series and round to the third decimal place. However, the books' answer uses the fourth term and rounds to the fourth decimal place. Using the remainder estimation theorem may not be helpful in this case as it requires knowledge of the (n+1)th derivative of the function.
frozenguy

Homework Statement

Any problem. Ex Use the Mac series of $$tan^{-1}\left(x\right)$$ to approximate at $$x=\frac{1}{2}$$ to three decimal place accuracy.

Homework Equations

$$\sum^{\infty}_{k=0}(-1)^{k}\frac{x^{2k+1}}{2k+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-...$$

The Attempt at a Solution

I found out using the remainder estimation theorem that all I need is up to the fifth power. So I put x into the expanded series to the fifth power (third term) and then I take three decimal places, rounding the third.
The books answer is four decimal places, fourth rounded and an answer that comes from using the seventh power (fourth term).

So, what am I doing wrong.. I have encountered this on several problems in the book.
Am I using my remainder estimation theorem wrong?

$$\frac{\left(\frac{1}{2}\right)^{n+1}}{n+1}<0.0005$$Using my calculator, I find at n=5, the left side is less then 0.0005

Anyone able to shed some light?

frozenguy - Just look at the terms of the series. At what point will the next sucessive term yield precision beyond your target. That's your cut-off point.

Using the Remainder Estimation therom requires knowledge of the (n+1)th derivative of f(x), something that might be very messy for your function, and will not (in this case) help you solve for the desired value of n (unless you try several trial-by-error cases).

Also, please recall/review, the Remainder Therom is

$$R_{n}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \cdot (x-a)^{(n+1)}$$

where $\xi=\xi(x)$ and here $a= 0$.

1. What is the purpose of approximating a mac/taylor series to z decimal accuracy?

The purpose of approximating a mac/taylor series to z decimal accuracy is to find an estimated value of a mathematical function at a specific point. This can be useful in situations where the exact value cannot be easily calculated or is too complicated to work with.

2. How is a mac/taylor series used to approximate a function?

A mac/taylor series is a way to represent a function as an infinite sum of terms. By taking a certain number of terms from the series and using them to calculate an estimated value, we can approximate the function at a specific point.

3. What does "z decimal accuracy" mean in the context of approximating a mac/taylor series?

"Z decimal accuracy" refers to the number of decimal places used in the approximation. For example, if we want to approximate a function to 3 decimal places, we would use the first 3 terms of the series to calculate the estimated value.

4. How do you determine the number of terms needed for a desired decimal accuracy?

The number of terms needed for a desired decimal accuracy can be determined by using a convergence test, such as the ratio test or the remainder test. These tests help us determine how many terms are needed to get the desired level of accuracy.

5. Are there any limitations to using a mac/taylor series to approximate a function?

Yes, there are limitations to using a mac/taylor series for approximation. The series may only converge for certain values of x, and may not accurately represent the function outside of its interval of convergence. Additionally, using too few terms can result in a large error in the approximation.

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