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Lecture 21 – Sequences

Explore arithmetic and geometric sequences, convergence, monotonicity, boundedness, series convergence, growth rates, and partial sums. Learn to determine convergence or divergence and calculate sums of various series.

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Lecture 21 – Sequences

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  1. Lecture 21 – Sequences A list of numbers following a certain pattern {an} = a1 , a2 , a3 , a4 , … , an , … Pattern is determined by position or by what has come before 3, 6, 12, 24, 48, …

  2. Defined by n(position) Find the first four terms and the 100th term for the following:

  3. Arithmetic Sequence An arithmetic sequence is the following: with a as the first term and d as the common difference.

  4. GeometricSequence A geometric sequence is the following: with a as the first term and r as the common ratio.

  5. Convergence We say the sequence “converges to L” or, if the sequence does not converge, we say the sequence “diverges”. A sequence that is monotonic and bounded converges.

  6. Monotonic and Bounded Monotonic: sequence is non-decreasing (non-increasing) Bounded: there is a lower bound m and upper bound M such that • Monotonic & Bounded: • Monotonic & not Bounded: • Not Monotonic & Bounded: • Not Monotonic & not Bounded:

  7. Example 1 – Converge/Diverge? Example 2 – Converge/Diverge?

  8. Lecture 22 – Sequences & Series Example 3 – Converge/Diverge? Growth Rates of Sequences: q, p > 0 and b > 1

  9. Example 4 – Converge/Diverge?

  10. Partial Sums Adding the first n terms of a sequence, the nth partial sum: Series – Infinite Sums If the sequence of partial sums converges, then the series converges.

  11. Find the first 4 partial sums and then the nth partial sum for the sequence defined by: Example 1

  12. The partial sum for a geometric sequence looks like: Geometric Series

  13. Lecture 23 – More Series Find the sum of the geometric series: Geometric Series – Examples

  14. Find the sum of the geometric series: Geometric Series – More Examples

  15. Telescoping Series – Example 1

  16. Telescoping Series – Example 1 – continued

  17. Telescoping Series – Example 2

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