Series

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Definition. is convergent'' or summable'' and if . Otherwise is divergent''.

Claim. When the right hand sides exist, so do the left hand sides and the equalities hold:

Claim. (The Boundedness Criterion) A nonnegative sequence is summable iff its partial sums are bounded.

Claim. (Cauchy's Criterion) is summable iff

Claim. (The Vanishing Condition) If is summable then .

Theorem 1. (The Comparison Test) If and converges, then so does .

Theorem 2. If and and then converges iff converges.

Theorem 3. (The Ratio Test) If and , then converges if and diverges if .

Theorem 4. (The Integral Test) If and is decreasing on and , then converges iff converges.

Definition. is absolutely convergent'' if converges.

Theorem 5. An absolutely convergent series is convergent. A series is absolutely convergent iff the series formed from its positive terms and its negative terms are both convergent.

Theorem 6. (Leibnitz's Theorem) If is non-increasing and then converges.

Theorem 7. (skipped) If converges but not absolutely, then for any number there is a rearrangement of so that .

Theorem 8. If converges absolutely and is a rearrangement of , then also converges absolutely and .

Theorem 9. If and converge absolutely and the sequence is composed of all the products of the form , then .

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Dror Bar-Natan 2003-03-17