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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13.5 – Sums of Infinite Series
Objectives: You should be able to
…Slide2
FormulasThe 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 FiniteSlide3
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
LimitsIf 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 sum
If 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