Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = e²ˣ, a = 0

Verified step by step guidance

1. Recall the Taylor series expansion of a function \( f(x) \) about \( a = 0 \) (Maclaurin series) is given by: \[ \displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
2. Identify the function \( f(x) = e^{2x} \). The derivatives of \( f(x) \) are all of the form \( f^{(n)}(x) = 2^n e^{2x} \). Evaluating at \( x = 0 \), we get: \[ f^{(n)}(0) = 2^n e^0 = 2^n \]
3. Substitute \( f^{(n)}(0) = 2^n \) into the Taylor series formula: \[ \displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n \]
4. Rewrite the power series in summation notation explicitly as: \[ \displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{(2x)^n}{n!} \]
5. This is the power series representation of \( e^{2x} \) centered at \( a = 0 \). The interval of convergence for this series is all real numbers \( (-\infty, \infty) \) because the exponential function's power series converges everywhere.

Key Concepts

Taylor Series Expansion

The Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (a). For f(x) = e^(2x) at a = 0, the series uses derivatives evaluated at 0 to express f(x) as a power series in (x - 0).

Power Series and Summation Notation

A power series is an infinite sum of terms in the form c_n(x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this series as Σ c_n (x - a)^n, making it easier to write and analyze the series representation of functions.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges to the function. Determining this interval involves testing the radius within which the infinite series converges, ensuring the series accurately represents the function within that range.