Explore the relationship between numerical sequences and Bernoulli's equality, uncovering the hereditary property that holds true for all natural integers in this in-depth analysis.
Extract
[...] - Conclusion: the property being true at the rank and hereditary, it is true for all natural integers. According to question A2) We deduce: Finally: On Thus, taking , on for all , (in fact the sequel is decreasing) Here is the algorithm to implement: as long as to do Return n In programming this algorithm on a calculator, we find We already know that the sequence is decreasing so for all : Let's show by recurrence on what is positive for all : - Initialisation : - Hereditary: suppose that at a certain rank we have So, then, as , on a So then: - Conclusion : the property is true at rank 0 and hereditary, it is therefore true for all integers The sequence being decreasing and minorated (by , she converges. [...]
[...] Numerical Sequences Part A In replacing part 1 in the Bernoulli's equality, we get: Or so In the previous question, we saw that . So we have Or according to the previous question Part B On a for We thus obtain the following table of variations: On The sequel so it is decreasing Let us show by induction that for all , : - Initialisation : Pour , so we have (in fact ) - Heredity: Suppose we have, at a certain level : So: As it is negative, we can deduce that Or on a sequence of inequalities: The property is therefore also true at the rank . [...]
