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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

…

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