A Cauchy Sequence is a type of mathematical sequence where the terms get closer and closer together as the sequence progresses. It is named after the French mathematician Augustin-Louis Cauchy.
To prove that a sequence is Cauchy, you must show that for any small positive number, there exists a point in the sequence after which all terms are within that distance from each other. This can be done by showing that the difference between any two terms in the sequence becomes smaller and smaller as the sequence progresses.
The formula for this particular Cauchy Sequence is a_n = [a_(n-1) + a_(n-2)]/2. This means that each term in the sequence is equal to the average of the two preceding terms. For example, if a_1 = 1 and a_2 = 2, then a_3 = (1+2)/2 = 1.5, a_4 = (2+1.5)/2 = 1.75, and so on.
This particular Cauchy Sequence has significance in the study of mathematical analysis and number theory. It is also used in various other areas of mathematics, such as in the proof of the Central Limit Theorem in statistics.
Yes, the Cauchy Sequence a_n = [a_(n-1) + a_(n-2)]/2 can converge to any number. This is because the sequence gets closer and closer together as it progresses, meaning that it can approach any number as its limit.