# 11.10 (Part 2) Power Series for e^x centered at 2 | Summary and Q&A

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December 6, 2018
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11.10 (Part 2) Power Series for e^x centered at 2

## TL;DR

This video explains how to find the power series expansion for e^x at two different centers using Taylor series.

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### 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!) * (x-a)^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.