Taylor Series Exploration

Taylor Series Exploration

Read each question carefully and provide detailed answers. Show all your work for full credit. Remember the Taylor series expansion formula: f(x) = Σ [fⁿ(a) / n!](x - a)ⁿ from n=0 to ∞.

1. A Taylor series is a representation of a function as an infinite sum of   calculated from the values of the function's derivatives at a single point.

2. When the Taylor series is centered at a = 0, it is called a   series.

3. The radius of convergence of a Taylor series determines the interval for which the series   to the function.

1. Which of the following functions is represented by the Taylor series expansion Σ (xⁿ / n!) centered at x = 0?

2. What is the value of 'a' when expanding a Maclaurin series?

1. Find the first three non-zero terms of the Taylor series for f(x) = sin(x) centered at a = π/2.

2. Determine the Taylor series for f(x) = eˣ centered at a = 1.

1. The Taylor series for a polynomial function is always the polynomial itself.

2. A Maclaurin series is a special case of a Taylor series where the series is centered at any point 'a'.

1. Use a Taylor series to approximate the value of √e to three decimal places. (Hint: Use the Taylor series for eˣ centered at a = 0 and evaluate at x = 0.5)