C2: Geometric Series Dr J Frost Last modified: 24 th September 2013.

Slides:

Advertisements

Similar presentations

C1 Chapter 6 Arithmetic Series Dr J Frost Last modified: 7 th October 2013.

Advertisements

Unit 12. Unit 12: Sequences and Series Vocabulary.

The sum of the infinite and finite geometric sequence

Assignment Answers: Find the partial sum of the following: 1. = 250/2 ( ) = 218, = 101/2 (1/2 – 73/4) = Find the indicated n th.

9-4 Sequences & Series. Basic Sequences  Observe patterns!  3, 6, 9, 12, 15  2, 4, 8, 16, 32, …, 2 k, …  {1/k: k = 1, 2, 3, …}  (a 1, a 2, a 3, …,

FP1 Chapter 6 – Proof By Induction

12-3 Infinite Sequences and Series. Hints to solve limits: 1)Rewrite fraction as sum of multiple fractions Hint: anytime you have a number on top,

Notes Over 11.3 Geometric Sequences

9.2 Arithmetic Sequence and Partial Sum Common Difference Finite Sum.

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,

Infinite Geometric Series

Sequences and Series A sequence is an ordered list of numbers where each term is obtained according to a fixed rule. A series, or progression, is a sum.

Section 11-1 Sequences and Series. Definitions A sequence is a set of numbers in a specific order 2, 7, 12, …

Aim: What are the arithmetic series and geometric series? Do Now: Find the sum of each of the following sequences: a)

Sequences/Series BOMLA LACYMATH SUMMER Overview * In this unit, we’ll be introduced to some very unique ways of displaying patterns of numbers known.

THE BEST CLASS EVER…ERRR…. PRE-CALCULUS Chapter 13 Final Exam Review.

Notes Over 11.4 Infinite Geometric Sequences

SEQUENCES AND SERIES Arithmetic. Definition A series is an indicated sum of the terms of a sequence.  Finite Sequence: 2, 6, 10, 14  Finite Series:2.

ADVANCED ALG/TRIG Chapter 11 – Sequences and Series.

Notes 9.4 – Sequences and Series. I. Sequences A.) A progression of numbers in a pattern. 1.) FINITE – A set number of terms 2.) INFINITE – Continues.

Lesson 4 - Summation Notation & Infinite Geometric Series

Sequences & Series. Sequences  A sequence is a function whose domain is the set of all positive integers.  The first term of a sequences is denoted.

Review of Sequences and Series.  Find the explicit and recursive formulas for the sequence:  -4, 1, 6, 11, 16, ….

Sequences, Series, and Sigma Notation. Find the next four terms of the following sequences 2, 7, 12, 17, … 2, 5, 10, 17, … 32, 16, 8, 4, …

Section 9.2 – Series and Convergence. Goals of Chapter 9.

Infinities 1 sequences and series. Sequence – an ordered set of numbers or other objects.

The Ratio Test: Let Section 10.5 – The Ratio and Root Tests be a positive series and.

SERIES: PART 1 Infinite Geometric Series. Progressions Arithmetic Geometric Trigonometric Harmonic Exponential.

AP Calculus Miss Battaglia  An infinite series (or just a series for short) is simply adding up the infinite number of terms of a sequence. Consider:

Series & Sequences Piecewise Functions

Infinite Series Copyright © Cengage Learning. All rights reserved.

Copyright © 2011 Pearson, Inc. 9.5 Series Goals: Use sigma notation to find the finite sums of terms in arithmetic and geometric sequences. Find sums of.

STROUD Worked examples and exercises are in the text Programme 11: Series 1 PROGRAMME 11 SERIES 1.

Sequences & Series: Arithmetic, Geometric, Infinite!

Chapter 6 – Sequence & Series The Questions in this revision are taken from the book so you will be able to find the answers in there.

10.2 Summing an Infinite Series Feb Do Now Write the following in summation notation 1, ¼, 1/9, 1/16.

Review of Sequences and Series

Geometric Sequence – a sequence of terms in which a common ratio (r) between any two successive terms is the same. (aka: Geometric Progression) Section.

13.3 Arithmetic and Geometric Series and Their Sums Finite Series.

Topic #1: Arithmetic and Geometric Sequences Objectives: Discuss the properties of functions Recognize Arithmetic Sequences and find a common difference.

9.3 Geometric Sequences and Series. Common Ratio In the sequence 2, 10, 50, 250, 1250, ….. Find the common ratio.

Copyright © 2007 Pearson Education, Inc. Slide Geometric Series A geometric series is the sum of the terms of a geometric sequence. Sum of the.

13.5 – Sums of Infinite Series Objectives: You should be able to…

S ECT. 9-2 SERIES. Series A series the sum of the terms of an infinite sequence Sigma: sum of.

JEOPARDY! $100 $100 $100 $100 $100 $200 $200 $200 $200 $200 $300 $300

Arithmetic and Geometric Sequences

The sum of the infinite and finite geometric sequence

nth or General Term of an Arithmetic Sequence

Geometric Sequences and Series (Section 8-3)

Infinite GP’s.

8.1 and 8.2 Summarized.

Aim: What is the geometric series ?

Infinite Geometric Series

Series & Sequences.

Geometric Series.

Infinite Geometric Series

Use summation notation to write the series

10.2 Arithmetic Sequences and Series

Find the sums of these geometric series:

Arithmetic Sequences:

Chapter 12 Review Each point your team earns is extra points added to your score on the upcoming test.

Calculus II (MAT 146) Dr. Day Monday, April 9, 2018

If the sequence of partial sums converges, the series converges

Objectives Find sums of infinite geometric series.

Warm Up Use summation notation to write the series for the specified number of terms …; n = 7.

Section 12.3 Geometric Sequences; Geometric Series

Presentation transcript:

C2: Geometric Series Dr J Frost Last modified: 24 th September 2013

Types of series 2, 5, 8, 11, 14, … +3 This is a: Arithmetic Series ? Geometric Series ? 3, 6, 12, 24, 48, … ? ?

Common Ratio ? ? ? ? ? ? ?

Common Ratio Exam Question May 2013 (Retracted) Hint for (a): the common ratio between the first and second terms, and the second and third terms, is the same. a b ? ?

Arithmetic SeriesGeometric Series Determine the following: 3, 6, 12, 24, … 40, -20, 10, -5, … ? ? ? ??

Another Common Ratio Example ?

Missing information The second term of a geometric sequence is 4 and the 4 th term is 8. The common ratio is positive. Find the exact values of: a)The common ratio. b)The first term. c)The 10 th term. The second term of a geometric sequence is 4 and the 4 th term is 8. The common ratio is positive. Find the exact values of: a)The common ratio. b)The first term. c)The 10 th term. ?

?

Exam Question Edexcel June 2010 ? ? ? ? ?

Arithmetic SeriesGeometric Series ? ? Technically you could be asked in an exam the proof of the sum of a geometric series (it once came up!) So let’s prove it…

Geometric Series Find the sum of the first 10 terms. ? ? ? ? ? ? ? ?

Summation Notation ? ? ? ?

Harder Questions: Type 1 ? ?

Exercise 7D ? ? ? ? ? ? ? ? ? ?

Different types of series What can you say about the sum of each series up to infinity? This is divergent – the sum of the values tends towards infinity. This is convergent – the sum of the values tends towards a fixed value, in this case 2. This is divergent. This is known as the Harmonic Series Just for fun... ? ? ? ? ?

Sum to Infinity ? ?

Examples ? ? ?? ? ? ? ? ? ? ? ?

A somewhat esoteric Futurama joke explained

Another Example ? ?

Edexcel May 2011 ? ? ? ?

Exercises Exercise 7D Q6, 7 Exercise 7E Q8 Exercise 7F Q10