Real analysis | limit of a sequence |ppt

B.sc 2nd semester , seminar, topic: limit of a sequence,

1. TOPIC: LIMIT OF A SEQUENCE

2. OUTLINES Introduction Limit of a sequence Some limit theorems Conclusion References

3. INTRODUCTION The concept of limits in sequences refers to the value that the terms approach as the index increases without bound. This idea of convergence is crucial for understanding the long-term behavior of sequences. Limit theorems, on the other hand, provide rules and properties that govern how limits behave in sequences, allowing us to make predictions and deductions confidently. Together, these concepts form the foundation for analyzing and solving problems across different mathematical and scientific fields .

4. LIMIT OF A SEQUENCE Definition : A sequence in is said to converge to ,or is said to be a limit of ,if for every there exists a natural number such that for all ,the terms satisfy . If a sequence has a limit ,we say that the sequence is convergent ; if it has no limit , we say that the sequence is divergent.

5. UNIQUENESS OF LIMITS Theorem(1 ) : A sequence in can have at most one limit. Proof : Suppose that and are both limits of . For each there exist such that Similarly , there exists such that Let , be the larger and . Then for we apply the Triangle inequility , to get

6. Since , is an arbitrary positive number , we conclude that

7. Theorem (2) : A convergent sequence of real number is bounded . Proof : Suppose that , And let . Then there exists a natural number such that Now , apply the triangle inequality , (here ) Let , is a bounded sequence .

8. Theorem (3) : If is a convergent sequence of real numbers and if for all then Proof : Suppose the conclusion is not true , i.e. Then is positive . Now, Since converges to . So there is a natural number such that In particular, But it contradicts the fact that

9. CONCLUSION Understanding the concept of limits in sequences is crucial for analyzing their behavior. Limits help determine convergence or divergence, and they have practical applications in fields like mathematics, physics, and engineering. Mastering this concept is key to tackling more complex mathematical topics.

10. REFERENCES Book : Introduction to real analysis (Forth Edition) Writer : Robert G. Bartle and Donald R. Sherbert Youtube : link : https://www.youtube.com/watch?v=cTnlHZD5ss4 Channel name : Wrath of Math

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