Adv. Analysis Test 2 Review

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10 Terms

1
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Convergence of a Sequence

A sequence (an) converges to a real number a if, for every positive number ε, there exists an N ∈ N such that whenever n ≥ N it follows that |an - a| < ε.

2
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Bounded Sequence

A sequence (xn) is bounded if there exists a number M > 0 such that |xn| ≤ M for all n ∈ N.

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Every convergent sequence is ________.

Bounded

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Increasing Sequence

A sequence (an) is increasing if an ≤ an+1 for all n ∈ N.

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Decreasing Sequence

A sequence (an) is decreasing if an ≥ an+1 for all n ∈ N.

6
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Monotone

A sequence is monotone if it is either increasing or decreasing.

7
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Monotone Convergence Theorem

If a sequence is monotone and bounded, then it converges.

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An increasing sequence of real numbers which is bounded above, ________.

Converges

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A decreasing sequence of real numbers which is bounded below, _________.

Converges

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Divergence of a Sequence

A sequence (an) will diverge if it does not converge.