|Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:||(252)||
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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 .