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

Series laws Ⅰ: finite series laws

1. Continuation. Let $m \leq n < p$ be integers, and let $a_{i}$ be a real number assigned to each integer $m \leq i \leq p$ . The we have

[

\sum_{i = m}^{n} a_{i} + \sum_{i = n + 1}^{p} a_{i} = ?

]

2. Indexing shift. Let $m \leq n$ be integers, $k$ be another integer, and let $a_{i}$ be a real number assigned to each integer $m \leq i \leq n$ . 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 $m \leq n$ be integers, and let $a_{i}, b_{i}$ be real numbers assigned to each integer $m \leq i \leq n$ . Then we have

\sum_{i = m}^{n} (a_{i} + b_{i}) = (\sum_{i = m}^{n} a_{i}) + (\sum_{i = m}^{n} b_{i})

4. [...]. Let $m \leq n$ be integers, and let $a_{i}$ be a real number assigned to each integer $m \leq i \leq n$ , and let $c$ be another real number. Then we have

\sum_{i = m}^{n} (c a_{i}) = c (\sum_{i = m}^{n} a_{i})

5. [...]. Let $m \leq n$ be integers, and let $a_{i}$ be a real number assigned to each integer $m \leq i \leq n$ . Then we have

| \sum_{i = m}^{n} a_{i} | \leq \sum_{i = m}^{n} | a_{i} |

6. [...]. Let $m \leq n$ be integers, and let $a_{i}, b_{i}$ be real numbers assigned to each integer $m \leq i \leq n$ . Suppose that $a_{i} \leq b_{i}$ for all $m \leq i \leq n$ . Then we have

\sum_{i = m}^{n} a_{i} \leq \sum_{i = m}^{n} b_{i}

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

07.12.2019