Question

The Taylor series for the function arcsin(x)arcsin⁡(x) about x=0x=0 is equal to ∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1.∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1. For this question,...

The Taylor series for the function arcsin(x)arcsin⁡(x) about x=0x=0 is equal to

∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1.∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1.

For this question, recall that 0!=10!=1.

a) (6 points) What is the radius of convergence of this Taylor series?

Write your final answer in a box.

b) (4 points) Let TT be a constant that is within the radius of convergence you found. Write a series expansion for the following integral, using the Taylor series that is given.

∫T0arcsin(x)dx∫0Tarcsin⁡(x)dx

Write your final answer in a box.

c) (5 points) Compute the following limit using Taylor Series.

limx→0arcsin(x)x.

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