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Warm UP!

Warm UP!. Identify the following as Arithmetic, Geometric, or neither: 2, 7, 12, 17, … 2. Find the nth term for the sequence: 2, 20, 200, 2000, … 3. Generate the first four terms of the sequence given its recursive formula: a 1 = 6 and a n = 2a n-1 + 3

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Warm UP!

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  1. Warm UP! Identify the following as Arithmetic, Geometric, or neither: 2, 7, 12, 17, … 2. Find the nth term for the sequence: 2, 20, 200, 2000, … 3. Generate the first four terms of the sequence given its recursive formula: a1 = 6 and an = 2an-1 + 3 4. Find the 150th term of the sequence an = 0.5n + 8 Quiz Time

  2. Graphical interpretation of limits for explicit sequences investigationComplete the task. You may work with a partner. You have 20 minutes. LG 6-2: Limits of Sequences

  3. A limit is like asking the following question: “What happens to a sequence an when n approaches infinity?” • A sequence can behave in two ways: • Convergence • Divergence

  4. Examples of converging sequences The sequence converges to a unique value: 0 We write this as

  5. Examples of converging sequences The sequence converges to a unique value: 1 We write this as

  6. Examples of diverging sequences This sequence diverges. You can see it is going up – to infinity. We can write this as

  7. Convergence or Divergence? Given a sequence an there are several possibilities as to its convergence behavior: • The sequence may converge to a number or to 0. • The sequence may diverge in one of three ways: • To positive infinity – “goes to infinity” • To negative infinity – “goes to negative infinity” • The limit might not exist at all (DNE)

  8. Limits for rational sequences • Converge • If the degrees are the same, the limit is equal to the ratio of the leading coefficients • If the degree of the denominator is larger than the numerator, the limit is 0. • Diverge • If the degree of the numerator is larger than the denominator, the limit goes to infinity

  9. Look at numbers 12 – 16 When a geometric sequence is in explicit form, you only need to use the common ratio to determine the limit as the sequence approaches infinity. The geometric sequence a1(r)n-1 is divergent if |r| > 1 or r > 1 and converges to 0 if |r| < 1.

  10. Limits Practice: 5. 8, -5, 8, -5, 8,. . . Estimate the limits: 7. 8.

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