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12.5 Infinite geometric Series

12.5 Infinite geometric Series. 9.2.2.4 Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. Guiding Question: How do I find the sum of something infinite?.

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12.5 Infinite geometric Series

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  1. 12.5 Infinite geometric Series 9.2.2.4 Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively

  2. Guiding Question: How do I find the sum of something infinite? • Lesson Objective: I will be able to find the sum of a infinite geometric series. I will be able to use mathematical induction to prove • Key terms: Converge, Limit, Diverge, mathematical induction, counterexample

  3. Guiding Question: How do I find the sum of something infinite? Consider this series When the common ratio (r) is a fraction r<1 What is the common ratio? What is the sum after each term (partial sum)? • This series is said to converge • Each partial sum will approach a number called a limit. • What is the limit of this series?

  4. Guiding Question: How do I find the sum of something infinite? Consider this series If common ratio is larger then 1 What is the common ratio? What is each partial sum? • The series approaches infinite and is said to diverge • The limit does not exist

  5. Which of these converge and which diverge? Guiding Question: How do I find the sum of something infinite?

  6. Sum of an infinite geometric series (if it exists) S = Sum Remember, this only works if a limit exists. Guiding Question: How do I find the sum of something infinite?

  7. To Prove a statement is true for all natural numbers n. • Show the statement true for n=1 • Assume the statement true for a natural number k. • Prove the statement true for (k+1) (the next term) For proof by mathematical induction I need to prove these 3 steps. Guiding Question: How do I find the sum of something infinite?

  8. Prove the sum of the first n odd numbers • Prove it is true for 1 • Assume it is true for k • Prove it is true for (k+1) • Add (k+1) to both sides and show that it is true. To Prove a statement is true for all natural numbers n. Show the statement true for n=1 Assume the statement true for a natural number k. Prove the statement true for (k+1) (the next term) Guiding Question: How do I find the sum of something infinite?

  9. Use mathematical induction to prove the following 1. To Prove a statement is true for all natural numbers n. Show the statement true for n=1 Assume the statement true for a natural number k. Prove the statement true for (k+1) (the next term) Guiding Question: How do I find the sum of something infinite?

  10. Guiding Question: How do I find the sum of something infinite? • You can also disprove a mathematical statement by using a counterexample • Find one case where the statement is false and you disproven it. • Find a counterexample

  11. Guiding Question: How do I find the sum of something infinite? • Assignment: Pg. 904 15-27 • Day 2 Pg. 904 30-38

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