Could I get some help with these series questions please? Thank you!!!

Question

b. use substituation and integration to find the first five nonzero terms for the maclaurin series for f(x) = sin^-1(x)

c. use both x=1 and x=1/2 and the answer in part b to find approximations for pi.

Answer 1

(1 + x)^-1/2 ≈ 1 + (-1/2)x + [(-1/2)(-3/2) / 2!]x² + [(-1/2)(-3/2)(-5/2) / 3!]x³ + [(-1/2)(-3/2)(-5/2)(-7/2) / 4!]x⁴

(1 + x)^-1/2 ≈ 1 - (1/2)x + (3/8)x² - (5/16)x³ + (35/128)x⁴

Replace x by -x²:

(1 - x²)^-1/2 ≈ 1 - (1/2)(-x²) + (3/8)(-x²)² - (5/16)(-x²)³ + (35/128)(-x²)⁴

(1 - x²)^-1/2 ≈ 1 + (1/2)x² + (3/8)x⁴ + (5/16)x⁶ + (35/128)x⁸

Integrate term by term:

Sin^-1x = ∫ (1 / √(1 - x²))dx ≈ x + (1/6)x³ + (3/40)x⁵ + (5/112)x⁷ + (35/1152)x⁹

So, since Sin^-1(1) = π/2, π/2 ≈ 1 + 1/6 + 3/40 + 5/112 + 35/1152 = 1.316691468

π ≈ 2(1.316691468) = 2.633382937

Since Sin^-1(1/2) = π/6, π/6 ≈ (1/2) + (1/6)(1/8) + (3/40)(1/32) + (5/112)(1/128) + (35/1152)(1/512)

π/6 ≈ (1/2) + (1/48) + (3/1280) + (5/14336) + (35/589824)

π/6 ≈ 0.523585195

π ≈ 6(0.523585195) = 3.141511172