Geometric Series

Core Sentence / 核心句

A geometric series has a constant ratio between consecutive terms.

中文理解：等比级数的后一项总是前一项乘同一个 ratio。

Formula / 公式或规则

$$\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r},|r|<1$$

When To Use / 什么时候用

看到 constant ratio、$(\text{constant}{)}^n$，先试 geometric series。

Watch Out / 易错点

$|r|\ge 1$ 发散；只有 $|r|<1$ 才能用求和公式。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第13次正课-无穷级数判敛 · Knowledge Point 3: Geometric Series Test
  • Knowledge Point 3: Geometric Series Test
• 微积分BC第13次正课-无穷级数判敛 · Example 2: Geometric Series 求和
  • Example 2: Geometric Series 求和
• 微积分BC第13次正课-无穷级数判敛 · 7. A geometric series has a constant ratio between consecutive terms.
  • 7. A geometric series has a constant ratio between consecutive terms.
• 微积分BC第14次正课-Ratio-Test与Taylor-Series · Knowledge Point 3: Choosing Among 7 Tests / 七种方法怎么选
  • | $\sum ar^n$ or constant ratio | Geometric Series |
• 微积分BC第13次正课-无穷级数判敛 · Knowledge Point 7: Comparison Test
  • - geometric series with $|r|<1$ converges
• 微积分BC第13次正课-无穷级数判敛 · Knowledge Point 8: Limit Comparison Test
  • 是 geometric series，converges。
• 微积分BC第13次正课-无穷级数判敛 · Example 7: Limit Comparison and Top-Degree Shortcut
  • 是 geometric series，ratio 是 $\frac12$，所以 converges。
• 微积分BC第13次正课-无穷级数判敛 · 8. Deep Phrases / 深句对应表
  • | A geometric series is the only early test that gives an exact sum. | 前几种判敛里，等比级数是少数能直接求和的。 |

Connected Notes / 连接

• 微积分BC第13次正课-无穷级数判敛
• Infinite Sequence and Series
• Ratio Test