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
.

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Dror Bar-Natan 2003-03-17