Question Detail
Question
Which of the following best suits the Taylor series expansion of a given function f(x) about the point x=b?
Correct Answer
Option A is the correct answer.
Detailed Explanation
The Taylor series of a real or complex-valued function f(x) that is infinitely differentiable at a real or complex number b is given by:f(x) = Σr=0∞ (f(r)(b) (x-b)r) / r!where f(r)(b) denotes the r-th derivative of f evaluated at the point b, and r! is the factorial of r.Option A perfectly matches this definition. Option B would be incorrect as the derivatives are evaluated at 0, but the expansion is about 'b'. Option C is the Maclaurin series (a Taylor series expanded around b=0). Option D is incorrect as it uses xr instead of (x-b)r.
Hint
Recall the general formula for a Taylor series expansion of a function f(x) about a point x=b. Pay attention to where the derivatives are evaluated and the term raised to the power of r.