Warm up
Download
1 / 13

Warm Up - PowerPoint PPT Presentation


  • 117 Views
  • Updated On :

Warm Up. Write the explicit formula for the series. Evaluate. Introduction to Series. What is a series? What does it mean for a series to converge? What are geometric and telescoping series? What is the nth term test?. Infinite Series. Partial Sums.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Warm Up' - bing


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Warm up l.jpg

Warm Up

Write the explicit formula for the series.

Evaluate.


Introduction to series l.jpg

Introduction to Series

What is a series?

What does it mean for a series to converge?

What are geometric and telescoping series?

What is the nth term test?




A series can either converge or diverge l.jpg
A series can either converge or diverge.

If the sequence of nth partial sums converges to A, then the series converges. The limit of S is called the sum of the series.

If the sequence of nth partial sums diverges, then the series diverges.


Ex find s 1 s 2 s 3 s 4 s 5 and an expression for s n l.jpg
Ex: Find S1, S2, S3, S4, S5, …and an expression for Sn

Therefore, the series converges to 1.

note: If the limit of the sequence is NOT 0, then the sum of the series must diverge.


This example was a geometric series because a n is an exponential function l.jpg
This example was a geometric series because an is an exponential function.

A geometric series with ratio (base) r

  • diverges if |r| > 1

  • Converges if 0 < r < 1

    If it converges then the series converges to



Find the sum of the series l.jpg
Find the sum of the series series.

This is called a telescoping series.


Find the sum of the series10 l.jpg
Find the sum of the series series.

This is another telescoping series.


Let s revisit the note we talked about before l.jpg
Let’s revisit the “note” we talked about before. series.

  • If the limit of the sequence is not 0, then the series diverges.

  • The contrapositive of this statement must also be true: If the series converges, then the limit of the sequence is 0.

  • However, the converse (and inverse) do not have to be true…and are NOT in this case. Just because the limit of the sequence is 0, the series can still diverge.

  • All of this information is classified as the “nth term test for divergence.” Always use this as your first step in answering the converge/diverge question for series.


In summary l.jpg
In Summary: series.

When asked if a series converges or diverges:

  • Do the nth term test for divergence.

  • If the series if geometric, find r and determine whether the series converges or diverges. If converges, find the sum.

  • If the series is a rational function and the denominator can be factored, separate the ratio by partial fractions and determine if it is a telescoping series. If it is, find the value to which it converges.


Slide13 l.jpg

Test series.

Series

Condition(s) for convergence

Conditions for divergence

Example & Comments


ad