Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.

Verified step by step guidance

1. Recall the definition of the sine integral function: \(\mathrm{Si}(x) = \int_0^x f(t) \, dt\), where \(f(t) = \frac{\sin t}{t}\) for \(t \neq 0\) and \(f(0) = 1\).
2. To find the Taylor series expansion of the integrand \(f(t)\) centered at 0, start with the Taylor series expansion of \(\sin t\) around 0: \(\sin t = \sum_{n=0}^\infty (-1)^n \frac{t^{2n+1}}{(2n+1)!}\).
3. Divide the series for \(\sin t\) by \(t\) to get the series for \(f(t) = \frac{\sin t}{t}\): \(f(t) = \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!}\).
4. Note that this series is valid for all \(t\) including \(t=0\) because the term for \(n=0\) is \(\frac{t^0}{1!} = 1\), which matches the given \(f(0) = 1\).
5. Thus, the Taylor series expansion of the integrand \(f(t)\) centered at 0 is \(f(t) = 1 - \frac{t^2}{3!} + \frac{t^4}{5!} - \frac{t^6}{7!} + \cdots\).

Key Concepts

Sine Integral Function

The sine integral function Si(x) is defined as the integral from 0 to x of (sin t)/t dt, with a special value at t = 0 to ensure continuity. It is important to understand this function as an example of an integral involving a non-elementary integrand that requires special handling at singular points.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually zero. Expanding the integrand (sin t)/t in a Taylor series centered at 0 involves expressing sin t as its power series and dividing by t, carefully handling the limit at t = 0.

Handling Indeterminate Forms and Continuity

Since (sin t)/t is undefined at t = 0, understanding limits and continuity is essential. Using the limit lim_{t→0} (sin t)/t = 1 ensures the function is well-defined and continuous at zero, which is crucial for correctly expanding the integrand in a Taylor series.