Question

Compute the Maclaurin series of the following functions using differentiation or integration of other known series. Indicate the domain where the expansion is valid. f(x) = ∫ˣ₀ e⁻ᵗ² dt.

Answer 1

The function f(x) is the integral of e to the negative square of t from 0 to x, and its Maclaurin series can be found by series manipulations. The validity of this series expansion is generally for all real numbers.

Step-by-step explanation:

The student asks about the computation of the Maclaurin series for the function f(x) = ∫⁴₀ e⁻¹² dt using differentiation or integration of other known series. This involves finding a power series expansion for a given function about the point x=0. The function f(x) represents the area under the curve of e⁻¹² from 0 to x, which does not have a simple elementary antiderivative. Therefore, the integration must be expressed as a series. The validity of the expansion will depend on the convergence of the series, which for entire functions like e⁻¹², typically extends to all real numbers. To find the Maclaurin series, one can differentiate or integrate a known power series expansion, in this case, perhaps starting with the series for e⁻¹.