11.10 (Part 2) Power Series for e^x centered at 2  Summary and Q&A
TL;DR
This video explains how to find the power series expansion for e^x at two different centers using Taylor series.
Questions & Answers
Q: What is the purpose of using Taylor series to find the power series expansion for e^x?
Taylor series is used because it provides a way to express a function as an infinite series of terms. This allows us to find the power series expansion for e^x.
Q: How do you determine the coefficients for the power series of e^x?
The coefficients can be found by differentiating e^x and evaluating the derivatives at the center of the series. Each derivative is divided by the corresponding factorial.
Q: What is the general formula for the power series expansion of e^x?
The power series expansion of e^x can be written as the sum from n=0 to infinity of (1/n!) * (xa)^n, where a is the center of the series.
Q: What is the significance of the radius of convergence for the power series of e^x?
The radius of convergence determines the range of x values for which the power series converges. In the case of e^x, the radius of convergence is infinity, meaning the series converges for all x values.
Summary & Key Takeaways

The video demonstrates how to find the power series expansion for e^x using Taylor series.

Differentiation is used to obtain the coefficients of the series.

The formula for the Taylor series is explained, along with the general formula for the power series of e^x.