Find the taylor series of the following atnbspxnbsp 0 to be

Question

1. Consider the following.

f(x) = x³e^{x^2}

Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of e^x.

2. Consider the following Taylor series expansion.

1/√(1+x) = 1 - (1/2)x + (1·3/2·4)x² - (1·3·5/2·4·6)x³ + (1·3·5·7/2·4·6·8)x⁴ -...

Find the Taylor series of the following at x = 0. (To be graded as correct, give your answer up to and including the x⁴ term. Answers with more or less terms will be graded as incorrect.)

f(x) = 1/√(1-x)

 

3. Consider the following Taylor series expansion.

1/√(1+x) = 1 - (1/2)x + (1·3/2·4)x² - (1·3·5/2·4·6)x³ + (1·3·5·7/2·4·6·8)x⁴ -...

Find the Taylor series of the following at x = 0. (To be graded as correct, give your answer up to and including the x⁸ term. Answers with more or less terms will be graded as incorrect.)

f(x) = 1/√(1-x²)