16.1數列

 

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16.1數列
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16-1 數列 (Sequence)

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一無窮數列(infinite sequence)……可表示成

定義1(數列Sequence極限之定義)

一數列有極限值(limit),則稱收斂至,寫成

假如存在,我們就稱此數列為收斂(the sequence converges)

反之,若一數列不收斂至,稱此數列為發散(the sequence diverges)

 

 

 

  

定理A

為收斂的兩個數列,且為一常數constant,則:

1.

2.

3.

4.

5.

6.

7.  if   and

 

 

 

1.     

 

解答:

      分子分母同除以

   

 

 

           

  

定理B

(夾擠定理.Squeeze Theorem)

,且

 

 

 

 

2.      證明

 

解答:

時,

 

由夾擠定理可得

 

  

定理C

,則

 

 

 

 

3. Evaluate if it exists.

 

 

解答:

由定理C

 

 

 

 

 

4.      Determine whether the sequence converge or diverge, and , if converges, find limit, where

 

解答:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.      Determine whether the sequence converge or diverge, and , if converges, find limit, where

 

解答:

So since converge to 0 by Squeeze Theorem

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6. Determine whether the sequence converge or diverge, and , if converges, find limit, where

 

解答:

So by the Squeeze Theorem and

converge to 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

解答:

 is decreasing since  

for each .

The Sequence is bounded since  for all . Note that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8. Determine whether the sequence converge or diverge, and , if converges, find limit, where

解答:

.As so

Convergence.

 

 

 

 

 

 

 

 

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