6.4 Series

6.4 Series

The sum of the terms of a sequence is a series. The $n$th partial sum is $S_n = a_1 + \cdots + a_n$.

For an arithmetic series:

Example 7

Sum of first $100$ positive integers: $S_{100} = 5050$.

Example 8

Series $2 + \dfrac{7}{2} + 5 + \dfrac{13}{2} + \cdots$: $d = \dfrac{3}{2}$, $a_{19} = 29$, $S_{19} = \dfrac{589}{2}$.

Example 9

$a_5 = 19$ and $S_4 = a_9 + 1$ $\Rightarrow$ $a_1 = 3$, $d = 4$; series $3 + 7 + 11 + \cdots$.

Example 10

How many terms of $-9 - 6 - 3 + 0 + \cdots$ sum to $66$? With $a_1 = -9$, $d = 3$: $n = 11$.

Example 11

$a_1 = 9$, $a_n = 105$, $S_n = 741$ $\Rightarrow$ $n = 13$, $d = 8$; first three terms $9, 17, 25$.