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

Series laws Ⅰ: finite series laws

1. Continuation. Let mn<p be integers, and let ai be a real number assigned to each integer mip. The we have

i=mnai+i=n+1pai=i=mpai

2. Indexing shift. Let mn be integers, k be another integer, and let ai be a real number assigned to each integer min. Then we have

i=mnai=j=m+kn+kajk

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

3. Linearity part I. Let mn be integers, and let ai,bi be real numbers assigned to each integer min. Then we have

i=mn(ai+bi)=(i=mnai)+(i=mnbi)

4. Linearity part II. Let mn be integers, and let ai be a real number assigned to each integer min, and let c be another real number. Then we have

i=mn(cai)=c(i=mnai)

5. Triangle inequality. Let mn be integers, and let ai be a real number assigned to each integer min. Then we have

|i=mnai|i=mn|ai|

6. Comparison test. Let mn be integers, and let ai,bi be real numbers assigned to each integer min. Suppose that aibi for all min. Then we have

i=mnaii=mnbi

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