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Math and science::Analysis::Tao::07. Series

Series laws Ⅰ: finite series laws

1. Continuation. Let \( m \le n < p \) be integers, and let \( a_i \) be a real number assigned to each integer \( m \le i \le p \). The we have

\[ \sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i = \sum_{i=m}^{p}a_i \]

2. Indexing shift. Let \( m \le n \) be integers, \( k \) be another integer, and let \( a_i \) be a real number assigned to each integer \( m \le i \le n \). Then we have

\[ \sum_{i=m}^{n}a_i = \sum_{j=m+k}^{n+k}a_{j-k} \]

This one above, I actually had trouble proving. It is almost too obvious.

3. Linearity part I. Let \( m \le n \) be integers, and let \( a_i, b_i \) be real numbers assigned to each integer \( m \le i \le n \). Then we have

\[ \sum_{i=m}^{n}(a_i + b_i) = \left( \sum_{i=m}^{n}a_i \right) + \left( \sum_{i=m}^{n}b_i \right) \]

4. Linearity part II. Let \( m \le n \) be integers, and let \( a_i \) be a real number assigned to each integer \( m \le i \le n \), and let \( c \) be another real number. Then we have

\[ \sum_{i=m}^{n}(c a_i) = c \left( \sum_{i=m}^{n}a_i \right) \]

5. Triangle inequality. Let \( m \le n \) be integers, and let \( a_i \) be a real number assigned to each integer \( m \le i \le n \). Then we have

\[ \left| \sum_{i=m}^{n}a_i \right| \le \sum_{i=m}^{n}|a_i| \]

6. Comparison test. Let \( m \le n \) be integers, and let \( a_i, b_i \) be real numbers assigned to each integer \( m \le i \le n \). Suppose that \( a_i \le b_i \) for all \( m \le i \le n \). Then we have

\[ \sum_{i=m}^{n}a_i \le \sum_{i=m}^{n}b_i \]

finite series → finite sets → infinite series → infinite sets (absolutely convergent series)


The 4 sets of series laws: part Ⅰ

Unique to finite series
The first two properties of finite series do not have parallels in any of the other 3 types of series. All other properties reappear for finite sets and infinite series.
Missing from absolutly converging series
For absolutely converging series, there is no triangle inequality defined as there is no definition (thus no value) for their conditional convergence. Tao doesn't define a comparison test for absolutely converging series either but I'm not sure why (maybe because it is covered by infinite series?)

Source

p157-158