Use Taylor’s Estimate to bound the error of the n-th order Maclaurin approximation of sin(a), where
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To derive a bound for the error in the n-th order Maclaurin approximation of sin(a), we can use Taylor's theorem. Taylor's theorem states that for a function f(x) that is infinitely differentiable on an interval containing x = a, the n-th order Taylor polynomial of f(x) centered at x = a is given by:
Pn(x) = f(a) + f'(a)(x-a) + (1/2!)f''(a)(x-a)^2 + ... + (1/n!)f^n(a)(x-a)^n
The error term, denoted by Rn(x), is the difference between the actual function value and the Taylor polynomial approximation:
Rn(x) = f(x) - Pn(x)
For sin(a), the Maclaurin series (Taylor series centered at x = 0) is given by:
sin(x) = x - (1/3!)x^3 + (1/5!)x^5 - (1/7!)x^7 + ...
To find a bound on the error of the n-th order Maclaurin approximation for sin(a), we need to analyze the behavior of the remainder term Rn(a) in the interval [a - R, a + R], where R is the positive radius of convergence of the Maclaurin series. In the case of sin(a), the Maclaurin series has an infinite radius of convergence for all real numbers, so we can consider any interval in the form [-R, R].
The remainder term Rn(a) can be expressed using the Lagrange form of the remainder (also known as the Peano form):
Rn(a) = (1/(n+1)!)f^(n+1)(c)(x-a)^(n+1)
where c lies between x and a. In our case, f(x) = sin(x). Therefore, we have:
Rn(a) = (1/(n+1)!)sin^(n+1)(c)(x-a)^(n+1)
To bound the error, we want to find an upper bound for |Rn(a)|. Since sin(x) can take values between -1 and 1, we have the following bound for |sin^(n+1)(c)|:
|sin^(n+1)(c)| <= 1
Therefore, we can simplify the bound for |Rn(a)| to:
|Rn(a)| <= (1/(n+1)!)|(x-a)^(n+1)|
Now we need to determine the maximum value of |(x-a)| in the interval [-R, R]. Since we have a = 0, the maximum value of |(x-a)| is equal to R.
Finally, using the bound for |Rn(a)| and the maximum value of |(x-a)|, we can derive the bound for the error of the n-th order Maclaurin approximation of sin(a):
|Rn(a)| <= (1/(n+1)!)(R)^(n+1)
This bound provides an estimate for the maximum error of the n-th order Maclaurin approximation of sin(a) in the interval [a - R, a + R].