warming up. 13.5: sums of infinite series pre-calculus
TRANSCRIPT
Warming Up
Find the sum of the first 5 terms:
First 6 terms. First 7 terms. If we continue what do you think will
happen to the sum???
13.5: Sums of Infinite Series
Pre-Calculus
Objectives:1. Evaluate Infinite Series2. Find interval of convergence3. Evaluate repeating decimals
Vocabulary:converge, diverge, interval of convergence
13.5 Sum of Infinite Series
Learning the Lingo/ Rules
Converge: To approach a limit Diverge: To fail to approach a limit
If , then the infinite geometric series converges to the sum
If , then the series diverges.
Example: Find the sum of the infinite geometric series: 9 – 6 + 4 – …
2 7
Quick Practice! Find the sum of the given infinite series,
otherwise state that it diverges!
1+3+9+27+…1+.1+.01+.001+…
10/9
Explain why the series converges or diverges. Show mathematical proof.
1 + 3 + 9 + 27 + …
For what values of x does the following infinite geometric series converge?
Solution (1 < x < 3) is called the “interval of convergence”
For what values of x does the following infinite series converge?
𝑥+𝑥2+𝑥3+…-1 < x < 1
The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. What is the
total vertical distance that the ball will travel before it stops?50
40
32
128/5
40
32
128/5
0S 2 5 500 50
504
15
450
An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest?
25
20
16
25
20
16
S
254
15
2 250
The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. What is the
total vertical distance that the ball will travel before it stops?
100
75
225/4
100
75
225/4
10S 80
100
4 43
1
0
10
3
Practice
TB P.502: 1-17 odd, 33, 34, 39, 40