Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: (252) Next: Homework Assignment 23
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Definition. $ \sum_{n=1}^\infty a_n$ is ``convergent'' or ``summable'' and $ \sum_{n=1}^\infty a_n = s$ if $ \sum_{n=1}^N a_n\to s$. Otherwise $ \sum_{n=1}^\infty a_n$ is ``divergent''.

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

$\displaystyle \sum_{n=1}^\infty (a_n+b_n) = \sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n,

$\displaystyle \sum_{n=1}^\infty c\cdot a_n = c\cdot\sum_{n=1}^\infty a_n. $

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

Claim. (Cauchy's Criterion) $ (a_n)$ is summable iff

$\displaystyle \lim_{n,m\to\infty} a_n+\dots+a_m = 0. $

Claim. (The Vanishing Condition) If $ (a_n)$ is summable then $ a_n\to 0$.

Theorem 1. (The Comparison Test) If $ 0\leq a_n\leq b_n$ and $ \sum b_n$ converges, then so does $ \sum a_n$.

Theorem 2. If $ a_n>0$ and $ b_n>0$ and $ \lim a_n/b_n=c\neq 0$ then $ \sum a_n$ converges iff $ \sum b_n$ converges.

Theorem 3. (The Ratio Test) If $ a_n>0$ and $ \lim
a_{n+1}/a_n=r$, then $ \sum a_n$ converges if $ r<1$ and diverges if $ r>1$.

Theorem 4. (The Integral Test) If $ f>0$ and $ f$ is decreasing on $ [1,\infty)$ and $ f(n)=a_n$, then $ \sum a_n$ converges iff $ \int_1^\infty f$ converges.

Definition. $ \sum_{n=1}^\infty a_n$ is ``absolutely convergent'' if $ \sum_{n=1}^\infty \vert a_n\vert$ 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 $ a_n$ is non-increasing and $ \lim a_n=0$ then $ \sum(-1)^n a_n$ converges.

Theorem 7. (skipped) If $ \sum a_n$ converges but not absolutely, then for any number $ s$ there is a rearrangement $ (b_n)$ of $ (a_n)$ so that $ s=\sum b_n$.

Theorem 8. If $ \sum a_n$ converges absolutely and $ (b_n)$ is a rearrangement of $ (a_n)$, then $ \sum b_n$ also converges absolutely and $ \sum a_n = \sum b_n$.

Theorem 9. If $ \sum a_n$ and $ \sum b_n$ converge absolutely and the sequence $ c_n$ is composed of all the products of the form $ a_ib_j$, then $ \sum c_n = \sum a_n \cdot \sum b_n$.

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Dror Bar-Natan 2003-03-17