417 1 suites theoreme de convergence

Takeaways

• 😀 The theorem on convergent sequences is introduced, focusing on the criteria that allow us to assert that a sequence converges.
• 😀 Sequences that are both increasing and bounded are convergent, meaning they approach a finite limit.
• 😀 A decreasing and bounded sequence is also convergent, following the same principle as increasing sequences.
• 😀 If an increasing sequence is not bounded, it tends toward positive infinity.
• 😀 The proof of the theorem starts by considering the set of values of the sequence and its upper bound, leading to the concept of a supremum.
• 😀 The limit of the sequence is shown to be the supremum of the set, demonstrating that an increasing sequence with an upper bound converges.
• 😀 An example with a sequence defined as 1 + 1/2 + 1/3 + ... shows that it is increasing and bounded, hence converges to a limit.
• 😀 Another example, the harmonic series, demonstrates that the sequence is increasing but not bounded, leading it to diverge to infinity.
• 😀 The harmonic sequence involves summing the inverses of natural numbers, and it is shown to grow without bound.
• 😀 The script concludes with a key takeaway about the behavior of the harmonic series, highlighting its tendency towards infinity.