1. In this question, we will see a technique commonly used in mathematics: comparison. We arcsin(x)...

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1. In this question, we will see a technique commonly used in mathematics: comparison. We arcsin(x) da) in two different ways and compare the result. In compute the same quantity: general, this method reveals some cool mathematics. In our case, we will indirectly show that n=1 converges and find the value it converges to. You can use the following two identities without proof: For any nonzero real number a, we have (1+x) = 1 + (a 1)(a 2) .. (a n + 1)). a(a-1)+ n=1 =1+x+ 2 n! a(a-1)(a-2) 3! for |x| < 1. (1) for |x| <1 12n+1 dx 12 2-4-6-8... (2n) = 1.3.5.7... (2n+1) (a) Use the identity (1) to show that 1 == =1+ n=1 1.3.5.7... (2n-1) 2n, for |x| <1. 2-4-6-8... (2n) (2) (b) Use the result in part (a) to find the Maclaurin series for arcsin(x) and determine its radius of convergence. (c) Use the Maclaurin series for arcsin(x) to express S arcsin(x) dr as an infinite power series. 2n+1 where cn is not (Hint: your series will have infinitely many terms of the form c a function of x.) arcsin(x) (d) In this step, we will evaluate dr using the power series we developed. With the help of the identity (2) show that arcsin(x) 1 dr = 1+ (2n-1) 32 2 n=1 1. In this question, we will see a technique commonly used in mathematics: comparison. We arcsin(x) da) in two different ways and compare the result. In compute the same quantity: general, this method reveals some cool mathematics. In our case, we will indirectly show that n=1 converges and find the value it converges to. You can use the following two identities without proof: For any nonzero real number a, we have (1+x) = 1 + (a 1)(a 2) .. (a n + 1)). a(a-1)+ n=1 =1+x+ 2 n! a(a-1)(a-2) 3! for |x| < 1. (1) for |x| <1 12n+1 dx 12 2-4-6-8... (2n) = 1.3.5.7... (2n+1) (a) Use the identity (1) to show that 1 == =1+ n=1 1.3.5.7... (2n-1) 2n, for |x| <1. 2-4-6-8... (2n) (2) (b) Use the result in part (a) to find the Maclaurin series for arcsin(x) and determine its radius of convergence. (c) Use the Maclaurin series for arcsin(x) to express S arcsin(x) dr as an infinite power series. 2n+1 where cn is not (Hint: your series will have infinitely many terms of the form c a function of x.) arcsin(x) (d) In this step, we will evaluate dr using the power series we developed. With the help of the identity (2) show that arcsin(x) 1 dr = 1+ (2n-1) 32 2 n=1