Sequences and Series 2

1
Sequences and Series 2

• Learn what a convergence sequence is
• Learn how to find the sum of the infinity in a
  geometric sequence

2
Geometric Series
We know from last lesson that the sum of the
terms in a geometric sequence can be given by
3
Example
Find the sum of the first 10 terms of 3 15 75
375
u1 3
r 5
n 10
4
Sum to Infinity

• Predicting what happens when the sum of a series
  gets larger

5
Investigating Series
Let Sn be the sum of first n terms of the
series 1 3 9 27 ..
What happens as n gets bigger??
6
Converging series

• A geometric series converges to a limit when..

Means the size of r including negatives
7
Converging series

• Which of the following values of r will converge
  to a limit?

0.9
0.005
-1.5
-0.03
-1
0.3
-0.2
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Sum to infinity

• The value to which a converging series
  approaches is called the sum to infinity.

1 0.5 0.25 0.125 0.0625 0.03125 .
What is the sum to infinity of this series??
9
Sum to infinity

• It is possible to work out the sum to infinity
  without just adding together the terms.

To do this we first need to understand what
happens to rn as n gets bigger, when r lt 0.
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Sum to infinity

• Lets look at our formula for geometric series
  again.

This can be replaced with 0. Why?
11
Sum to infinity

• How does this change our formula

This is the formula to find the sum to infinity
12
Example 1

• Find the sum to infinity of the geometric
  series

u11
r3/5
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Example 2

• The 1st term of a GS is -4 and the sum to
  Infinity is 9. Find the ratio of terms

u1 12
12 9(1-r)
12 9 - 9r
9r 9 - 12 -3
r - 1/3
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Example 3
Evaluate
r
u1 (1/2)0 1
1st term, when n0
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Different sorts of question to try
Sum to infinity
1 2/5 4/25
r
u1
2/5
1
sum 1 / 3/5 5/3
(1/3)11/3
r
u1
1/3
sum 1/3 / 2/3 1/2
As a fraction
r
u1
1/10
0.8 4/5
sum 4/5 / 9/10 8/9
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Activity
Turn to page 198 of your textbook and answer
questions in exercise 6E