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Warm UP!

Warm UP!. Find the limits: 1. 2. 3. Estimate the limits: 4. 5.

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Warm UP!

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  1. Warm UP! Find the limits: 1. 2. 3. Estimate the limits: 4. 5.

  2. A group of thirty explorers discover a money tree with 3 dollar bills on it. The explorers decide they will share the money from the tree equally. Every day the tree grows an additional 3 bills. But, every day two more people hear about the tree and show up to claim their share and all the money is redistributed so each person has the same amount. • Write the explicit formula for the sequence giving the amount of money, , that each person has on the nth day. • Estimate the limit of the amount of money each person will get.

  3. A pile driver pounds a piling into the ground for a new building that is being constructed. Suppose that on the first impact the piling is driven 100cm into the ground. On the second impact the piling is driven another 80 cm into the ground. Assume the distance the piling is driven with each impact forms a geometric sequence. • How far will the piling be driven on the tenth impact? • How deep in the ground will the piling be after 10 impacts? • Find the limit of the sequence. • Find the limit of the sum.

  4. Investigation Consider the series: 1 + ½ + ¼ + … Make a table of the first 10 sums of this series. What do you notice? Consider a similar series: 1 + 2 + 4 + …. Again, make a table. How is this table similar or different from the one above? Why do think this is so?

  5. Of the two series, which would have an infinite sum? Explain. • There is a formula to find the limit of a series that is convergent: • Find, using the formula above, the sum of the convergent infinite geometric series.

  6. State whether the geometric series converges. If it does, find the limit of the series. • 100 + 90 + 81 + … • 25 + 20 + 16 + … • 40 + 50 + 62.5 + … • 200 – 140 + 98 - …

  7. Math Induction and Partial Sums • Induction is a technique used for proving a statement is true of EVERY value of n (all integers). • For example: Given the series , find the explicit formula for the nth partial sum. • Use mathematical induction to prove your formula is correct.

  8. Math Induction and Partial Sums • Given the series , find the explicit formula for the nth partial sum. • Use mathematical induction to prove your formula is correct.

  9. Math Induction and Partial Sums • Given the series , find the explicit formula for the nth partial sum. • Use mathematical induction to prove your formula is correct.

  10. Math Induction and Partial Sums • Given the series , find the explicit formula for the nth partial sum. • Use mathematical induction to prove your formula is correct.

  11. Practice

  12. Questions? • Please STUDY for your test NEXT CLASS! • You need to know • How to tell if a sequence is arithmetic or geometric • Find the common difference or ratio • Write recursive formulas for both • Solve for the nth term • Find limits of sequences AND series • Find partial sums • Induction HOMEWORK: pg 661 # 23 – 26 and # 33 – 36; pg 665 T1-T6, T8, T9, T15, T16

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