13.6 Limits of Sequences. We have looked at sequences, writing them out, summing them, etc. But, now let’s examine what they “go to” as n gets larger.

Slides:

Advertisements

Similar presentations

Series Slides A review of convergence tests Roxanne M. Byrne University of Colorado at Denver.

Advertisements

Series Brought to you by Tutorial Services – The Math Center.

Knight’s Charge  Quiz #1  When you finish your quiz, pick up the “INIFINITE GEO SERIES” worksheet and begin working on it! NOTE: The original segment.

Theorems on divergent sequences. Theorem 1 If the sequence is increasing and not bounded from above then it diverges to +∞. Illustration =

Determine whether the sequence 6, 18, 54, is geometric. If it is geometric, find the common ratio. Choose the answer from the following :

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,

Convergence or Divergence of Infinite Series

Alternating Sequences. Definition A sequence is said to be alternating or oscillating if it is divergent, but it neither diverges to infinity or to minus.

Notes Over 11.3 Geometric Sequences

A sequence is geometric if the ratios of consecutive terms are the same. That means if each term is found by multiplying the preceding term by the same.

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,

13.7 Sums of Infinite Series. The sum of an infinite series of numbers (or infinite sum) is defined to be the limit of its associated sequence of partial.

Wednesday, March 7 How can we use arithmetic sequences and series?

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

9.1 Sequences. A sequence is a list of numbers written in an explicit order. n th term Any real-valued function with domain a subset of the positive integers.

+ Geometric Sequences & Series EQ: How do we analyze geometric sequences & series? M2S Unit 5a: Day 9.

Goal: Does a series converge or diverge? Lecture 24 – Divergence Test 1 Divergence Test (If a series converges, then sequence converges to 0.)

Warm Up Finish your open notes quiz from yesterday. When you come in, I will return your quiz to you. Remember, you may use your 4 pages of notes and the.

Notes Over 11.2 Arithmetic Sequences An arithmetic sequence has a common difference between consecutive terms. The sum of the first n terms of an arithmetic.

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

Alternating Series.

9.5 Part 1 Ratio and Root Tests

Lesson 11-1 Sequences. Vocabulary Sequence – a list of numbers written in a definite order { a 1, a 2, a 3, a n-1, a n } Fibonacci sequence – a recursively.

Acc. Coordinate Algebra / Geometry A Day 36

Infinite Geometric Series

Lesson 11-2 Series. Vocabulary Series – summation of a infinite sequence ∑ s 1 + s 2 + s 3 + s 4 + ….. + s n Partial Sum – sum of part of a infinite sequence.

Sequences (Sec.11.2) A sequence is an infinite list of numbers

What you really need to know! A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same.

CHAPTER Continuity Series Definition: Given a series   n=1 a n = a 1 + a 2 + a 3 + …, let s n denote its nth partial sum: s n =  n i=1 a i = a.

Lesson 11-5 Alternating Series. Another Series Type Alternating Series – a series of numbers that alternate in sign, like the summation of the following.

Sequences Lesson 8.1. Definition A succession of numbers Listed according to a given prescription or rule Typically written as a 1, a 2, … a n Often shortened.

Warm Up  Name the geometric figures below and identify whether they are defined or undefined.

In this section, we will begin investigating infinite sums. We will look at some general ideas, but then focus on one specific type of series.

Geometric Series. In a geometric sequence, the ratio between consecutive terms is constant. The ratio is called the common ratio. Ex. 5, 15, 45, 135,...

9.1 Power Series Quick Review What you’ll learn about Geometric Series Representing Functions by Series Differentiation and Integration Identifying.

Series A series is the sum of the terms of a sequence.

Review of Sequences and Series

Thursday, March 8 How can we use geometric 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.

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

11-5 Geometric Series Hubarth Algebra II. A geometric series is the expression for the sum of the terms of a geometric sequence. As with arithmetic series,

Ch. 10 – Infinite Series 9.1 – Sequences. Sequences Infinite sequence = a function whose domain is the set of positive integers a 1, a 2, …, a n are the.

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…

Copyright © Cengage Learning. All rights reserved Series.

Lesson 69 – Ratio Test & Comparison Tests

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

Series and Convergence (9.2)

The sum of the first n terms of an arithmetic series is:

nth or General Term of an Arithmetic Sequence

Lesson 13 – 6 Limits of Sequences

Infinite Geometric Series

Lesson 64 – Ratio Test & Comparison Tests

Convergence or Divergence of Infinite Series

Pre Calculus 11 Section 1.5 Infinite Geometric Series

Math –Series.

Homework Questions.

Both series are divergent. A is divergent, B is convergent.

If the sequence of partial sums converges, the series converges

11-6. Convergent Geometric Series

Determine whether the sequence converges or diverges. {image}

Power Series, Geometric

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

Packet #29 Arithmetic and Geometric Sequences

Lesson 11-4 Comparison Tests.

Homework Questions.

Section 12.3 Geometric Sequences; Geometric Series

Presentation transcript:

13.6 Limits of Sequences

We have looked at sequences, writing them out, summing them, etc. But, now let’s examine what they “go to” as n gets larger and larger without bound. Consider What are the terms going to? Pick a bigger number. Looks like they get close to 0. So, we say *Note: There is no term that IS 0, it just gets SUPER CLOSE to 0!

To find the limit of a sequence, we usually have to algebraically manipulate it. Ex 1) Find the limit of a sequence as n increases without bound. a) b) goes to 0

If a sequence gets closer to a number, L, as n increases without bound, it is said to converge and it is a convergent sequence. If it does not converge, the sequence is said to diverge. *Note: If it does diverge, it can do so by several ways – getting larger & larger or by oscillating. Ex 2) Find the first 5 terms and decide if the sequence converges or diverges. a) b) getting larger  diverges diverges

Ex 3) Determine whether these geometric sequences converge or diverge. a) b) c) d) diverge diverge converge converge Can you generalize these geometric sequences & make a rule for what will converge & what will diverge? Try On Your Own! Ex 4) Characterize each sequence as convergent or divergent. If it converges, give the limit. 0 convergent divergent n

Homework #1306 Pg 718 #1–16 all, 18, 20, 22–27 all, 30, 32, 34–36