本章大綱 9.1 Sequence數列 9.2 Infinite Series無窮級數 9.3 Tests for Convergence審斂法 9.4 Power Series, Maclaurin Series, Taylor Series冪級數、 麥可勞林級數和泰勒級數 9.5 Taylor and Maclaurin Series 泰勒級數與麥克勞林級數

9.1.1 Sequence 數列 例：

9.1.2 The Limit of a Sequence數列的極限

9.1.3 Limit laws for sequences

9.1.4 Theorem 例題1. Calculate . 解：

9.1.5 The squeeze theorem for sequence 夾擠定理 例題2. Discuss the convergence of the sequence . 解：

例題3. For what values of is the sequence convergent. 解：

9.1.6 Monotonic Sequences單調數列

9.1.7 Bounded Sequence 有界數列

例題4. Show that the sequence is convergent. 證明：

9.1.8 Theorems

證明：

例題5. Find . 解：

9.1.9四個重要的極限 證明：

9.1.10 Theorem 例題6. convergent or divergent. 解： Determine whether the sequence is convergent or divergent. 解：

9.2 Infinite Series 無窮級數 9.2.1 Series

例題1. Geometrie Series 幾何級數 . 解：

例題2. Find the sum of the geometric series . 解：

例題3. Is the series convergent or divergent? 解：

例題4. Write the number as a ratio of integers. 解：

例題5. Telescoping Series疊進級數 解：

例題6. Harmonic Series調和級數 解：

9.2.2 Theorem

9.2.3 Theorem 級數收斂的必要條件： 證明：

9.2.4 The Test for Divergence 例題7. 利用 test 知 解：

例題8. 例題8. Find the sum of the series . Find the sum of the series . 解：

9.3 Tests for Convergence審斂法 9.3.1 The Integral Test 積分審斂法

證明：

例題1. Prove that p-series is convergent if and divergent if 證明：

例題2. Determine whether the series converges or diverges. 解：

9.3.2 The Comparison Test比較審斂法

證明： 註：

例題3. 解： Determine whether the series converges or diverges. (1) (2) (3) 解：

9.3.3 The Limit Comparison Test 極限比較審斂法

證明：

例題4. Test the series for convergence or divergence. (1) (2) 解：

註：

9.3.4 The Ratio Test 比值審斂法

證明：

例題5. Test the series for convergence or divergence. (1) (2) 解：

9.3.5 The Root Test 根值審斂法

例題6. 解： Which of the following series converges, and which diverges? (1) (2) 解：

9.3.6 The Alternating Series Test 交錯級數審斂法

證明：

例題7.

例題8. Test for convergence or divergence. 解：

9.3.7 Absolutely Convergent 絕對收斂 證明：

註：

例題9. Test the series for convergence or divergence. (1) (2) 解：

9.4 Power Series, Maclaurin Series, Taylor Series 冪級數、麥可勞林級數和泰勒級數 冪級數不但是形式簡單，而且在其收斂區間內可以像對多項式一般對它進行各種運算。因此，將一個函數表示成冪級數，對於研究函數的性質有著重要的意義。

9.4.1 Power Series 冪級數

例題1. 解： For what values of x do the following power series converge ? (1) (2) (3) 解：

9.4.2 The Convergence Theorem for Power Series 證明：

9.4.3 Radius of Convergence收斂半徑

例題2. 解： Find the radius and the interval of convergence of the series. (1) (2) (3) 解：

例題3. 解： Find the radius and the interval of convergence of the series. (1) (2) 解：

9.5 Taylor and Maclaurin Series 泰勒級數與麥克勞林級數 9.5.1 The Representation of a Function as a Power Series

例題1. Find a power series representation for the function. (3) 解：

Theorem

例題2. Find a power series representation for the function 解：

9.5.2 Taylor series and Maclaurin series

例題3. 解：

例題4. Find the Taylor series of and its radius of convergence. 解：

9.5.3 Taylor’s Theorem 泰勒定理 註：

Taylor’s Formula 泰勒公式

證明：

9.5.4 The Remainder Estimation Theorem

例題5. Find the Maclaurin series for and its radius of convergence. 解：

例題6. Find the Maclaurin series of and its radius of convergence. 說明：

解：

例題7. Find the Maclaurin series of and its radius of convergence. 說明：

解：

例題8. Find the Maclaurin series of and its radius of convergence. 解：

9.5.5 泰勒餘式定理的應用 例題9. 解：

Evaluate correct to within an error of 0.001. 例題10. Evaluate correct to within an error of 0.001. 解：