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Question 4.T.2: (Cauchy’s Criterion) The series ∑xn is convergent if and onl...

(Cauchy’s Criterion)

The series \sum{x_{n}} is convergent if and only if, for every ε > 0 there is a positive integer N = N (ε) such that

n > m ≥ N ⇒ |S_{n} − S_{m}| = |x_{m+1} + · · · + x_{n}| < ε.        (4.1)

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This is a direct application of the Cauchy criterion to the sequence S_{n} =∑^{n}_{k=1} x_{k}.

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