Sequences and Series

Test - April 19
Series: the sum of the terms of a sequence
Arithmetic
- $t_{n} = a + (n - 1) d$
- series $S_{n} = \frac{n (2 a + (n - 1) d)}{2} = \frac{n (a + t_{n})}{2}$
Geometric
- $t_{n} = a r^{n - 1}$
- series $S_{n} = \frac{a (1 - r^{n})}{1 - r} = \frac{a - r t_{n}}{1 - r}$ ( $r \neq 1$ )
- r can be 0 but mostly acceptably when $a \neq 0$
- infinite geometric series
  - when | r | < 1, $S = \frac{a}{1 - r}$
  - when | r | $\geq$ 1, the series has no finite sum
Sigma notation
- $\sum_{k = 1}^{n} 2 k^{2}$ read as "the sum of $2 k^{2}$ as k goes from 1 to n"