Sum of the Convergent Infinite Series presentation.pptx

Sum of the Convergent Infinite Series

1. Recap: Tell whether the geometric series is finite or infinite: • • • of a series that starts with 2 and multiplies each term by 3, ending at 162. •  

2. Recap: Identify the common ratio, then tell whether the series is divergent or convergent, : • •  

3. Recap: • What is the difference between finite and infinite geometric series? • What is the difference between a convergent and divergent series? • When is an infinite geometric series convergent ? Answer:   Sum approaches to a finite number The series is decreasing. Sum does not approach to a finite number and increasing. countable Not countable

4. Convergent Infinite Geometric Series

5. On the first day , you eat 2 whole pizzas . On the second day, you eat 1 whole pizza . Since you don’t know how long you’ll survive, you decide to eat wisely, so on the third day, you eat only 1/2 of a pizza . On the fourth day, you eat 1/4 of pizza , and this continues — each day you eat half as much as the previous day.” “The Convergence of Survival”

6. Here are the Questions: “If this continues forever, how much pizza will you have eaten in total?” “Will it be just 3 pizzas? Less than 4 pizzas? Exactly 4 pizzas? Or more than 4 pizzas?”

7. Concrete Phase: Guide Questions What happens to the pizza portions each day? Does the total seem to grow endlessly, or does it approach a certain number? What do you predict will be the total if the process never ends?

8. Pictorial Phase: Guide Questions How does the drawing or illustration compare to the pizza slices? What do the shaded or occupied area seem to approach?

9. Abstract Phase: Guide Questions What is the formula for the convergent infinite geometric series?

10. Activity (5 minutes) : State if the series is convergent or divergent. a. b. c. If convergent, find the sum.  

11. Real-life example: “A bouncing ball reaches half the previous height every bounce. What's the total distance it travels, if the ball is dropped 6 feet above the ground?”

12. Real-life example:

13. Generalizations: When can we find the sum of an infinite geometric series ? Why is it important that the ratio is less than 1? How does the formula help us find totals quickly?

14. Evaluation (5 minutes): Determine if the series converges. b. c. If convergent, find the sum.  

15. Thank you 