Unit 10B Review

The Taylor series expansion is $$2x + 2x^2 + \frac{8x^3}{3!} + \frac{16x^4}{4!} + ...$$ .

The radius of convergence is 2.

The interval of convergence is $$-\infty < x < \infty$$ .

The series represents $$\sin e$$ .

The fourth term is $$-\frac{x^{39}}{7!}$$ .

A power series is an infinite series of the form $$\sum_{n=0}^{\infty} a_n (x - c)^n$$ , where $$a_n$$ are coefficients and $$c$$ is a constant.

The Alternating Series Test states that if the terms of a series alternate in sign and decrease in absolute value to zero, then the series converges.

The radius of convergence determines the interval within which the power series converges.

The Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

The Taylor series expansion is $$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$ .