Objectives: You should be able to. …. Formulas. The goal in this section is to find the sum of an infinite geometric series. However, this objective is very closely connected to the limit of an infinite sequence. . ID: 585412
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Presentations text content in 13.5 – Sums of Infinite Series
Slide1
13.5 – Sums of Infinite Series
Objectives: You should be able to
…
Slide2
Formulas
The goal in this section is to find the sum of an infinite geometric series. However, this objective is very closely connected to the limit of an infinite sequence. Compare the sum of infinite geometric series to that of a finite series. Infinite Finite
Slide3
How did we get there?
Consider the following sequence of partial sums: using the finite geom. formula simplified
Slide4
Continued..
Now consider the limit of . Since the sequence of partial sums has a limit of 1, we say that the infinite series has a sum of 1 as well.
Slide5
Limits
If the infinite sequence of partial sums ( ) has: A finite limit, then it converges to the sum of S An infinite (approaches infinity or no limit) it is said to diverge.
Slide6
Also noted:
If , the infinite geometric series converges to the sumIf and , then the series diverges.
Slide7
Example:
Find the first three terms of an infinite geometric sequence with sum 16 and common ratio .
Slide8
Example:
Show that the series is geometric and converges to if , where n is an integer.
Slide9
Example:
The infinite, repeating decimal 0.4545454545….. can be written as the infinite series 0.45 + 0.0045 + 0.000045 + ….What is the sum of this series?
Slide10
Example:
What is the sum of the series for 5.363636… ?
Slide11
INTERVAL OF CONVERGENCE
Ex. Find the a) interval of convergence, b) the sum
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DownloadNote - The PPT/PDF document "13.5 – Sums of Infinite Series" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Presentations text content in 13.5 – Sums of Infinite Series
13.5 – Sums of Infinite Series
Objectives: You should be able to
…
Slide2Formulas
The goal in this section is to find the sum of an infinite geometric series. However, this objective is very closely connected to the limit of an infinite sequence. Compare the sum of infinite geometric series to that of a finite series. Infinite Finite
Slide3How did we get there?
Consider the following sequence of partial sums: using the finite geom. formula simplified
Slide4Continued..
Now consider the limit of . Since the sequence of partial sums has a limit of 1, we say that the infinite series has a sum of 1 as well.
Slide5Limits
If the infinite sequence of partial sums ( ) has: A finite limit, then it converges to the sum of S An infinite (approaches infinity or no limit) it is said to diverge.
Slide6Also noted:
If , the infinite geometric series converges to the sumIf and , then the series diverges.
Slide7Example:
Find the first three terms of an infinite geometric sequence with sum 16 and common ratio .
Slide8Example:
Show that the series is geometric and converges to if , where n is an integer.
Slide9Example:
The infinite, repeating decimal 0.4545454545….. can be written as the infinite series 0.45 + 0.0045 + 0.000045 + ….What is the sum of this series?
Slide10Example:
What is the sum of the series for 5.363636… ?
Slide11INTERVAL OF CONVERGENCE
Ex. Find the a) interval of convergence, b) the sum