Power Series

Core Sentence / 核心句

A power series is a series whose terms contain powers of x around a center.

中文理解：幂级数是带变量的级数；它不是只问数值收敛，还要问哪些 $x$ 让它收敛。

Formula / 公式或规则

$$\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$$

When To Use / 什么时候用

看到 series in $x$、centered at $c$、radius/interval of convergence，就用 power series framework。

Watch Out / 易错点

不要只对 $n$ 做判敛，忘了最后要解出 $x$ 的范围。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• Chapter 11 · Infinite Power-Series Form
  • a Taylor series is a power series whose coefficients come from derivatives at the center
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Knowledge Point 20: Power Series Depends on x
  • Knowledge Point 20: Power Series Depends on x
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Knowledge Point 17: Series Can Be Differentiated and Integrated
  • The power series is equal to the function on its interval of convergence.
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Knowledge Point 21: Use Ratio Test First
  • 对 power series 先用 Ratio Test。
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Knowledge Point 8: Taylor’s Big Idea
  • All smooth functions can be represented by power series near a point.
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · 1. What This Lecture Is About / 这节课在讲什么
  • Part B: start Taylor / Maclaurin / Power Series
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · 2. Link to Previous Lecture / 和第13次正课的连接
  • 最后进入 power series，因为 power series 的收敛区间还是要靠 Ratio Test。
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · 9. Deep Phrases / 深句对应表
  • | Taylor turns functions into power series. | 复杂函数变成幂函数，方便求导积分。 |
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Convergence Test Map
  • Factorials / exponentials / powers / power series?

Connected Notes / 连接

• 微积分BC第14次正课-Ratio-Test与Taylor-Series
• Taylor Series
• Maclaurin Series
• Interval of Convergence
• 11 - Taylor Series
• Taylor Polynomial
• Radius of Convergence