How do i calculate sqrt n i


Next page:Limits of functions and Upwards:Completeness of the real numbers Previous page:Monotonous consequences & nbsp content

nth roots

Comment: As a starting value you can e.g. choose. Then .

Proof. The estimates follow by induction .

The first two statements are clear by definition.
There follows after Bernoulli ():
.
   .
   .
So it exists . From the recursion formula it follows:
   .
Hence is .

Description. The uniquely determined number from the previous sentence is called the -th root . Description: One sets .

Proof.

Uniqueness:
Be there . If , then . Out so follows .
Existence:
The existence of the nth root follows from the determination .
Theorem 2.2.17 (Bernoulli's inequality for the root)

For , , and applies:

   .

Proof. We sit

.
Then . According to Bernoulli () follows
   
   

If we apply the inequality just shown, it follows:
.

Proof. The case is clear. If the limit so there is a so that

For .
For .
The claim now follows from Bernoulli's inequality:
   .
Finding 2.2.19 Be it , . Then
.
The consequence
is

Comment: The convergence follows from Bernoulli's inequality: For applies:

   .

Example.

   ,    
   .    

Proof. For put one

With
and apply Bernoulli's inequality at:
   .
So is .

In the event of is and from

the strict monotony follows the sequence: .

In the event of are the reciprocal values strictly monotonically falling.

Finding 2.2.20 The consequence , ( ), is strictly decreasing and it is

Comment. The claims follow from the estimate

For

Proof. According to Lemma applies

For .
For .
We sit .
   .
So is
   .


Next page:Limits of functions and Upwards:Completeness of the real numbers Previous page:Monotonous episodes & nbsp content Analysis1-A. Lambert 2001-02-09