Find the first 3 terms in the Maclaurin series of√(1-x+x^2)

Question

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Answer 1

The MacLaurin Series for y = f(x) is f(0) + f'(0)(x-0) + f"(0)/2! (x-0)² + f'''(0)/3! (x-0)³ + ...

 

f(x) = √(1-x+x²)      f(0) = 1

 

f'(x) = (1/2)(1-x+x²)^-1/2(2x-1)    f'(0) = -1/2

 

f"(x) = (1/2)[2(1-x+x²)^-1/2 + (2x-1)(-1/2)(1-x+x²)^-3/2(2x-1)]      f"(0) = 3/4

 

Sum of first 3 terms of MacLaurin Series:

 

   f(0) + f'(0)x +( f"(0)/2!) x² = 1 - (1/2)x + (3/8)x²