Lecture 17 – 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, …

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

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

Geometric Sequence A geometric sequence is the following: 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.

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:

Example 1 – Converge/Diverge?

Lecture 18 – Sequences & Series Example 3 – Converge/Diverge? Growth Rates of Sequences: q, p > 0 and b > 1

Example 4 – Converge/Diverge?

Partial Sums Series – Infinite 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.

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

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

Lecture 19 – More Series Geometric Series – Examples Find the sum of the geometric series:

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

Telescoping Series – Example 1

Telescoping Series – Example 1 – continued

Telescoping Series – Example 2

Lecture 20 – Divergence Test Goal: Does a series converge or diverge? Divergence Test (If a series converges, then sequence converges to 0.)

Example 1 – Converge/Diverge?

Example 3 – Converge/Diverge? However,

Example 4 – Converge/Diverge? However,

Lecture 21 – Integral Test Goal: Does a series converge or diverge? Integral Test (The area under a function and infinite sum of the terms in a sequence defined by that function are related.)

If area under curve is bounded, then so is But then is a bounded, monotonic sequence. So it converges and thus converges. If area under curve is unbounded, then is also unbounded. And thus, diverges.

Example 1 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 2 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 3 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Lecture 22 – Ratio and Root Tests Goal: Does a series of positive terms converge or diverge? Ratio Test (Does ratio of successive terms approach some limit L? Then series is close to being geometric.)

Root Test (Does nth root of terms approach some limit L Root Test (Does nth root of terms approach some limit L? Then series is close to being geometric.)

Example 1 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 2 For what values k does the series converge?

Example 3 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge?

Lecture 23 – Comparison Tests Direct Comparison: 1. 2.

Limit Comparison:

Example 1 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 2 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 3 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge? sequence converges to zero. No info from Divergence Test.

Lecture 24 – Alternating Series Test Goal: Does a series (of terms that alternate between positive and negative) converge or diverge?

Alternating Series Test A series of the form with is called alternating and converges if:

Example 1 – Converge/Diverge?

Example 2 – Converge/Diverge?

Example 3 – Converge/Diverge?

Example 4 – Converge/Diverge?

Using Sn (nth partial sum) to approximate S The remainder term when using Sn:

Example 5 How many terms are needed to approximate the series and yield a remainder less than 0.0001?

Absolutely Converge, Conditionally Converge, or Diverge

Example 6 – AC, CC, D? Look at absolute value:

Example 7 – AC, CC, D? Look at absolute value:

Example 8 – AC, CC, D? Look at absolute value: