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

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

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

$\sum_{i=m}^{n}a_i = \sum_{j=m+k}^{n+k}[...]$

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

3. [...]. 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. [...]. 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. [...]. 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. [...]. 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)