Question

A function f has derivatives of all orders for −1(d) Let Pₙ(1/2) represent the nth-degree Taylor polynomial for g about x = 0 evaluated at x = 1/2, where g is the function defined in Part C. Use the alternating series error bound to show that |P₄(1/2)-g(1/2)|<(1/500).

a) |P₄(1/2)-g(1/2)| < 1/500
b) |P₄(1/2)-g(1/2)| > 1/500
c) |P₄(1/2)-g(1/2)| = 1/500
d) Cannot be determined

Answer 1

To use the alternating series error bound to show that |P₄(1/2)-g(1/2)|<1/500, we can start by expressing the alternating series error bound formula: |Rₙ(x)| <= aₙ₊₁, where Rₙ(x) represents the remainder term of the nth-degree Taylor polynomial, and aₙ₊₁ is the absolute value of the (n+1)th term of the series. Since the alternating series is not given, it's not possible to determine the exact value of a₄₊₁, so the option d) Cannot be determined is the correct answer.

Step-by-step explanation:

To use the alternating series error bound to show that |P₄(1/2)-g(1/2)|<1/500, we can start by expressing the alternating series error bound formula:

|Rₙ(x)| <= aₙ₊₁, where Rₙ(x) represents the remainder term of the nth-degree Taylor polynomial, and aₙ₊₁ is the absolute value of the (n+1)th term of the series.

In this case, we have the fourth-degree Taylor polynomial P₄(1/2), so we need to find the absolute value of the fifth term of the series, which is a₄₊₁.

Since the alternating series is not given, it's not possible to determine the exact value of a₄₊₁, so the option d) Cannot be determined is the correct answer.