Chapter 9.2, Problem 29E

To determine

The terms of the Maclaurin series for $\cos x$ that are needed to approximate the value of $\cos 18$ accurate to with $0.001$ of the true value.

Answer to Problem 29E

The given series is converges.

Explanation of Solution

Given Information:

The term for series is defined as,

  $\cos 18$

Consider the given series,

  $\cos 18$

It is known that $\cos 18\approx 0.6603$ .

The series is defined as,

  ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}\frac{x^{2n}}{\left(2n\right)!}$

Now, use the graphing calculator and the obtained sum is $0.6582$ for $N=25$ . Other hand the obtained sum is $0.6606$ for $N=26$ , which is within $0.001$ of the obtained exact value. As the calculation start form $N=0$ . So the total required term is 27.

Hence, the required term is 27.