# How do i calculate sqrt n i

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### 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

**.**

**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

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**content**Analysis1-A. Lambert 2001-02-09

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