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Limits of Sequences of Real Numbers

Limits of Sequences of Real Numbers. Sequences of Real Numbers Limits through Rigorous Definitions The Squeeze Theorem Using the Squeeze Theorem Monotonous Sequences. Sequences of Numbers. Definition. 1. Examples. 2. 3. Limits of Sequences. Definition.

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Limits of Sequences of Real Numbers

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  1. Limits of Sequences of Real Numbers Sequences of Real NumbersLimits through Rigorous DefinitionsThe Squeeze Theorem Using the Squeeze Theorem Monotonous Sequences

  2. Sequences of Numbers Definition 1 Examples 2 3

  3. Limits of Sequences Definition If a sequence has a finite limit, then we say that the sequence is convergent or that it converges. Otherwise it diverges and is divergent. Examples 1 0

  4. 0

  5. Limits of Sequences 2 The sequence (1,-2,3,-4,…) diverges. 3 Notation

  6. Computing Limits of Sequences (1)

  7. 1 0 n2 Computing Limits of Sequences (1) Examples 1 2

  8. Computing Limits of Sequences Examples continued 3

  9. Formal Definition of Limits of Sequences Definition Example

  10. Limit of Sums Theorem Proof

  11. By the Triangle Inequality Limit of Sums Proof

  12. Limits of Products The same argument as for sums can be used to prove the following result. Theorem Remark Examples

  13. Squeeze Theorem for Sequences Theorem Proof

  14. Here each term k/n < 1. Using the Squeeze Theorem Example Solution This is difficult to compute using the standard methods because n! is defined only if n is a natural number. So the values of the sequence in question are not given by an elementary function to which we could apply tricks like L’Hospital’s Rule.

  15. Using the Squeeze Theorem Problem Solution

  16. Monotonous Sequences A sequence (a1,a2,a3,…) is increasing if an ≤ an+1 for all n. Definition The sequence (a1,a2,a3,…) is decreasing if an+1 ≤ an for all n. The sequence (a1,a2,a3,…) is monotonous if it is either increasing or decreasing. The sequence (a1,a2,a3,…) is bounded if there are numbers M and m such that m ≤ an ≤ M for all n. A bounded monotonous sequence always has a finite limit. Theorem Observe that it suffices to show that the theorem for increasing sequences (an) since if (an) is decreasing, then consider the increasing sequence (-an).

  17. Monotonous Sequences A bounded monotonous sequence always has a finite limit. Theorem Let (a1,a2,a3,…) be an increasing bounded sequence. Proof Then the set {a1,a2,a3,…} is bounded from the above. By the fact that the set of real numbers is complete, s=sup {a1,a2,a3,…} is finite. Claim

  18. Monotonous Sequences A bounded monotonous sequence always has a finite limit. Theorem Let (a1,a2,a3,…) be an increasing bounded sequence. Proof Let s=sup {a1,a2,a3,…}. Claim Proof of the Claim

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