The period of a pendulum with length that makes a maximum angle with the vertical is

Chapter 11, Problem 34

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The period of a pendulum with length that makes a maximum angle with the vertical is where and is the acceleration due to gravity. (In Exercise 40 in Section 7.7 we approximated this integral using Simpsons Rule.) (a) Expand the integrand as a binomial series and use the result of Exercise 44 in Section 7.1 to show that T 2 L t 1 12 22 k 2 12 32 22 42 k 4 12 32 52 22 42 62 k 6 k If is not too large, the approximation ,obtained by using only the first term in the series, is oftenused. A better approximation is obtained by using twoterms:(b) Notice that all the terms in the series after the first one havecoefficients that are at most . Use this fact to compare thisseries with a geometric series and show that(c) Use the inequalities in part (b) to estimate the period of apendulum with meter and . How does it comparewith the estimate ? What if ?