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Matt B.
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Matt B. said:Our definition of convergence provided is the following:
{an} converges to A if ∀ε>0, ∃a positive integer N such that n ≥ N => | an - A | < ε.
I'm not sure how I am supposed to use this in the example above?
Yes.Ray Vickson said:Have you ever heard of Cauchy sequences?
Matt B. said:Yes.
Ray Vickson said:OK... so?
Matt B. said:Do I just assume that xn can be substituted for the usual xn and that xn+1 be substituted for xm and attempt to find the n,n+1 ≥ N such that | xn - xn+1 | < ε, given ε > 0?
A sequence converges if its terms approach a specific value (called the limit) as the number of terms in the sequence increases. This means that the terms get closer and closer to the limit value, and eventually stay within a certain distance (called the epsilon value) from the limit value.
To prove that a sequence converges, you must show that the terms of the sequence get arbitrarily close to the limit value. This can be done through various methods, such as the epsilon-delta definition, the squeeze theorem, or the monotone convergence theorem.
The epsilon-delta definition of convergence states that a sequence converges to a limit L if, for any epsilon value (ε) greater than 0, there exists a corresponding delta value (δ) such that if the distance between any term in the sequence and L is less than δ, then the distance between all subsequent terms and L will also be less than ε.
No, a sequence can only have one limit. If a sequence has multiple limit points, it is considered to be divergent.
Yes, the limit of a convergent sequence is unique. This means that if a sequence has a limit, that limit value will be the same regardless of the method used to prove convergence.