A list of numbers following a certain pattern { a n } = a 1, a 2, a 3, a 4, …, a n, … Pattern is determined by position or by what has come before 3, 6, 12, 24, 48, … Lecture 21 – Sequences 1

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

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

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

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. 5

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

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

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

Example 4 – Converge/Diverge? 9

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

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

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

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

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

15 Telescoping Series – Example 1

16 Telescoping Series – Example 1 – continued

Telescoping Series – Example 2 17

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