Newton's Binomial Theorem to Estimate, Find Error

[jillian3]

Homework Statement

Use Newton's Binomial Theorem to estimate integral of (1+x^4)^(1/2) from 0 to 1/2 to within one part in 1000, (error>1/1000)

Homework Equations

I used the Binomial Series expansion, so (a+b)^n = a^n +na^(n-1)b + (n(n-1))/2! (etc

The Attempt at a Solution

I understand how to do a binomial series expansion, but I don't know how to use it to estimate a value and find the error in this case. I'm using the series expansion to estimate a value for integral given. So I need to add up several of the functions in the series expansion, subtract it from the value for the integral of (1+x^4)^(1/2) from 0 to 1/2. But the value of this is not an exact number, and I can't put it in a format of (1+x) and plug in for my expansion to get values.

First off, is this how you use the theorum to estimate? Is that how you find the error? And if that is the case, how d I find the error or estimate with the information I'm given?

 

[mighty2000]

Thank you for your question! Using the Binomial Theorem to estimate a value for an integral is a valid approach, but there are a few steps you need to take to find the error and ensure your estimate is within one part in 1000.

First, you will need to rewrite the integral in terms of (1+x)^n, where n is a positive integer. In this case, we can rewrite (1+x^4)^(1/2) as (1+(x^4/4))^2, which allows us to use the Binomial Theorem.

Next, we can use the Binomial Theorem to expand (1+(x^4/4))^2 to get an infinite series. However, to ensure that our estimate is within one part in 1000, we only need to add up a certain number of terms in the series. This number of terms can be determined using the error formula for Taylor series, which is given by |R_n(x)| ≤ M(x-a)^(n+1)/(n+1)!, where R_n(x) is the remainder term after adding n terms of the series, M is the maximum value of the (n+1)th derivative of the function on the interval (0, 1/2), and a is the center of the series expansion. In this case, a = 0, and we can find the maximum value of the (n+1)th derivative by taking the derivative of (1+x^4)^(1/2) and evaluating it at x = 1/2.

Once you have determined the appropriate number of terms to add in the series, you can plug in x = 1/2 to get an estimate for the integral. To find the error, you can use the error formula mentioned above and plug in the appropriate values for M, n, and x.

I hope this helps! Let me know if you have any further questions.Scientist