Chapter 11.4, Problem 78E

Sine integral function The function $\text{Si(}x\text{) = }{\displaystyle \underset{0}{\overset{x}{\int }}f(t)dt}$, where $f(t)=\{\begin{array}{cc}\frac{\sin t}{t}& \text{if }t\ne 0\\ 1& \text{if }t=0\end{array}$, is called the sine integral function.

1. a. Expand the integrand in a Taylor series centered at 0.
2. b. Integrate the series to find a Taylor series for Si.
3. c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed 10^-3.