Integrated Math 3 / Unit 9: Sequences, Series and Functions Review
9.3 Series and Sigma Notation
Notes
Sigma notation $\displaystyle \sum_{k=1}^{n} a_k$ means to add the terms $a_1$ through $a_n$.
- Example: $\displaystyle \sum_{k=1}^{5} (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35$.
- An infinite geometric series converges only if $|r| < 1$, and then $\displaystyle S = \frac{a_1}{1 - r}$.
- Example: $\displaystyle 8 + 4 + 2 + 1 + \dots = \frac{8}{1 - \frac{1}{2}} = 16$.
- If $|r| \geq 1$, the infinite series does not converge.
Practice
Evaluate $\displaystyle \sum_{k=1}^{5} (2k + 1)$.
The terms are $3, 5, 7, 9, 11$ and they add to 35.
What is the sum of the infinite series $8 + 4 + 2 + 1 + \dots$?
$\displaystyle r = \frac{1}{2}$, so $\displaystyle S = \frac{8}{1 - \frac{1}{2}} = 16$.
A ball is dropped from 10 feet and each bounce reaches 60% of the previous height. About how far does it travel in total before stopping?
Down 10, then each bounce goes up and down: $\displaystyle 10 + 2\left(\frac{6}{1 - 0.6}\right) = 10 + 2(15) = 40$.
Ready for more? Take the Unit 9 test.