1. 9.1 Part 1
Sequences and Series

2. Write a general expression for the following sequential terms:
This is called an arithmetic sequence
As n goes to infinity, what does an approach?

3. 1
Does the sum of the terms: 1
approach a limiting value?
Use the 1 X 1 square above to find your answer.

4. You can start with this one by one square:1
You can start with this one by one square: 1
Since the area of all of these slices of the square add up to the area of the square…
This is an example of an infinite series.
Show the solution here.
Note: In the future, could we use this same approach to show that the harmonic series diverges?
This series converges (approaches a limiting value.)

5. A ball bounces with elasticity such that each time it bounces
it returns to half it’s previous height. If the ball was dropped
from a height of 5 feet, find the total distance that the ball
travels.
Note to improve this problem: We might try making this a ball that is sprung up from the ground so that the first 5 feet are doubled along with the other terms later in the series.

6. 5 A ball bounces with elasticity such that each time it bounces
it returns to half it’s previous height. If the ball was dropped
from a height of 5 feet, find the total distance that the ball
travels. 5 …
At this point, I expect iterations to be done on the calculator and for students to notice that each iteration gets the approximation closer to 15.
What seems to be the answer you are all getting? 5
We’ll come back to this…is there a pattern that you are seeing here?

7. a1, a2,… are terms of the series. an is the nth term.
In an infinite series:
a1, a2,… are terms of the series.
an is the nth term.
Partial sums:
nth partial sum
If Sn has a limit as , then the series converges, otherwise it diverges.

8. Geometric Series:
In a geometric series, each term is found by multiplying the preceding term by the same number, r. Now we want to see if we can find a short formula for
Subtract the second line from the first. What do you get?
Can you determine the limit from this formula?

9. The partial sum of a geometric series is:
If then

10. Geometric Series:
In a geometric series, each term is found by multiplying the preceding term by the same number, r.
This converges to if , and diverges if
is the interval of convergence.

11. Geometric Series:
In a geometric series, each term is found by multiplying the preceding term by the same number, r.
This converges to if , and diverges if
A geometric series can be expressed as
As long as the first exponent is of r is 0 or

12. Example 1: a r

13. Example 2: a r

14. p 5 Now let’s return to the bouncing ball that bounces
to half it’s previous height each time it bounces. If the ball
was dropped from a height of 5 feet, find the total distance
that the ball travels. 5 p