# 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

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

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

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

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

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

*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?)