# Series, Convergence, Divergence

## Lecture Notes

#### Sequences

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Section 1, Page 1 to page 2

Definition, with examples of convergent and divergent sequences.

Instructor: Prof. Jason Starr
Prior Knowledge: Limits (section 2 of lecture 2)

#### Tests for Convergence/Divergence of Sequences

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Section 2, Page 2 to page 3

The squeezing lemma and the monotone convergence test for sequences.

Instructor: Prof. Jason Starr
Prior Knowledge: Sequences (section 1 of this lecture)

#### Series

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Section 3, Page 3 to page 4

Definition, using the sequence of partial sums and the sequence of partial absolute sums. Absolutely convergent and conditionally convergent series are defined, with examples of the harmonic and alternating harmonic series.

Instructor: Prof. Jason Starr
Prior Knowledge: Sequences (section 1 of this lecture)

## Online Textbook Chapters

#### Introduction to Infinite Series

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Definitions of sequences and series, with examples of harmonic, geometric, and exponential series as well as a definition of convergence.

Prior Knowledge: None

#### Manipulating Absolutely Convergent Series

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Examples of the uses of manipulating or rearranging the terms of an absolutely convergent series.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Infinite Series (OT30.1)

#### Computing Series Partial Sums

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Steps for using a spreadsheet to compute the partial sums of a series.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Infinite Series (OT30.1)

## Exam Questions

#### Infinite Series

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Problem 17 (page 2)

Determining whether a given series converges or diverges.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Infinite Series

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Problem 6C-1 (page 41) to problem 6C-3 (page 41)

Three questions which involve finding the sum of a geometric series, writing infinite decimals as the quotient of integers, determining whether fifteen different series converge or diverge, and using Riemann sums to show a bound on the series of sums of 1/n.

Instructor: Prof. David Jerison
Prior Knowledge: None