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If 0<r<1 and |x(n+1) - x(n)| < r ^n for all n. Show that x(n) is Cauchy sequence.
help please.
help please.
A Cauchy Sequence is a sequence of numbers in which the elements become arbitrarily close to each other as the sequence progresses.
0 To show that x(n) is a Cauchy Sequence, we need to prove that for any given positive number ε, there exists a positive integer N such that for all n,m ≥ N, the absolute value of the difference between x(n) and x(m) is less than ε. Proving that x(n) is a Cauchy Sequence is important because it is a necessary condition for a sequence to be considered convergent. It also helps us understand the behavior of a sequence and make predictions about its future elements. Yes, apart from the 0How do you show that x(n) is a Cauchy Sequence?
What is the importance of proving that x(n) is a Cauchy Sequence?
Are there any other conditions that need to be met for x(n) to be considered a Cauchy Sequence?
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