Which expression correctly represents the nth-degree Taylor polynomial of f about a?

Which expression correctly represents the nth-degree Taylor polynomial of f about a?

• A. f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^{(n)}(a)(x-a)^n/n!
• B. f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^{(n-1)}(a)(x-a)^{n-1}/(n-1)!
• C. f(a) + f'(a)x + f''(a)x^2/2! + ... + f^{(n)}(a)x^n/n!
• D. f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^{(n+1)}(a)(x-a)^{n+1}/(n+1)!

Answer: (A)

The expression represents the Taylor polynomial of degree n centered at a. It builds the polynomial by taking derivatives of f at a and forming terms that involve powers of (x − a) scaled by the factorial of the derivative order. Writing out the terms from k = 0 to n gives f(a) + f'(a)(x − a) + f''(a)(x − a)^2/2! + ... + f^{(n)}(a)(x − a)^n/n!. This is the standard truncation of the Taylor series at degree n, using each derivative up to the nth and the shifted variable (x − a). The other forms either stop too early (missing the nth term), replace (x − a) with x (which would be a Maclaurin expansion about 0, not about a), or extend to the (n+1)th derivative, giving degree n+1. The exact form above is the one that matches a degree-n Taylor polynomial about a.

The expression represents the Taylor polynomial of degree n centered at a. It builds the polynomial by taking derivatives of f at a and forming terms that involve powers of (x − a) scaled by the factorial of the derivative order. Writing out the terms from k = 0 to n gives f(a) + f'(a)(x − a) + f''(a)(x − a)^2/2! + ... + f^{(n)}(a)(x − a)^n/n!. This is the standard truncation of the Taylor series at degree n, using each derivative up to the nth and the shifted variable (x − a).

The other forms either stop too early (missing the nth term), replace (x − a) with x (which would be a Maclaurin expansion about 0, not about a), or extend to the (n+1)th derivative, giving degree n+1. The exact form above is the one that matches a degree-n Taylor polynomial about a.