Series laws Ⅰ: finite series laws
1. Continuation. Let
2. Indexing shift. Let
This one above, I actually had trouble proving. It is almost too obvious.
3. Linearity part I. Let
4. Linearity part II. Let
5. Triangle inequality. Let
6. Comparison test. Let
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?)