Find the taylor series atnbspxnbsp 0 of the given function

Question

1. Consider the following.

f(x) = 1 / (1 + x)

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 1/(1 - x). For reference: This series is derived in your textbook. (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.)

2. Consider the following.

f(x) = x / (1 - x)³

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 1/(1 - x). For reference: This series is derived in your textbook. (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.)

3. Consider the following.

∫1 / (1 + x³) dx

Find the Taylor series expansion at x = 0 of the given antiderivative. (To be graded as correct, give your answer up to and including the x¹⁰ term. Answers with more or less terms will be graded as incorrect.)

4. Consider the following.

∫ x e^{x^3} dx

Find the Taylor series expansion at x = 0 of the given antiderivative. (To be graded as correct, give your answer up to and including the x¹¹ term. Answers with more or less terms will be graded as incorrect.)