Question

Compute the Maclaurin series for the following function F(x)=∫₀ˣ​ 1/4+t² What are its radius and interval of convergence? What is a more common name for F(x) ? (For example, a 'more common name' for the function ∫ˣ₁​ t/1​ dt is ln(x). )

Answer 1

The Maclaurin series for F(x) = ∫₀ˣ (1/4+t²) is 0, and its radius of convergence is infinity. A more common name for F(x) is the antiderivative or indefinite integral of (1/4+t²).

Step-by-step explanation:

To find the Maclaurin series for the function F(x) = ∫₀ˣ (1/4+t²) dt, we can begin by integrating the function. The integral of 1/4+t² with respect to t is (1/4)t + (1/3)t³, evaluated from 0 to x. This simplifies to (1/4)x + (1/3)x³. Since we are finding the Maclaurin series, we can replace x with 0 to get the general term of the series, which is 0.

Next, we can find the radius of convergence by considering the values of x for which the Maclaurin series converges. In this case, the radius of convergence is infinity, because the series has a constant term and all higher degree terms are 0. Therefore, the series converges for all real values of x.

A more common name for F(x) = ∫₀ˣ (1/4+t²) dt is the antiderivative function or the indefinite integral of (1/4+t²).