1. Geometric Sequences & Series This week the focus is on convergent series and finding the sum of a convergent geometric series.

2. Geometric Sequences & Series CONTENTS: What is a convergent series? Finding the Sum to Infinity Example 1 Example 2 Example 3 Assignment

3. Geometric Sequences & Series What is a convergent series? A convergent series occurs when the common ratio for a series is between -1 and +1. If -1 < r < 1, each term is getting smaller and smaller and therefore the sum of the series approaches a number but will never exceed that number. The number the sum approaches is called the Sum to Inifinity.

4. Geometric Sequences & Series Finding the Sum to Infinity: The Sum to Inifinity is found using the following formula: where a is the first term, r is the common ratio and ∞ is the symbol for infinity.

5. Geometric Sequences & Series Example 1: Find the sum to infinity of: 1 + 0.1 + 0.01 + 0.001 + … Solution: a = 1r = 0.1/1 = 0.1 Substitute these values into the formula to get:

6. Geometric Sequences & Series Example 2: Find the common ratio of a geometric series with first term 10 and sum to infinity of 30. Solution: a = 10r = ? Substitute the values we have into the formula to create and equation: Cross multiply Expand brackets

7. Geometric Sequences & Series Example 3: The sum of the first three terms of a geometric series is 9 and its sum to infinity is 8. What could you deduce about the common ratio? Find the first term and the common ratio. Solution: (i)As the sum of the first three terms is bigger than the sum to infinity this would suggest that the common ratio for the series is a negative number. continued on next slide

8. Geometric Sequences & Series (ii) From the question we are told: S 3 = 9 and S ∞ = 8. We can create two equations from this using the formulae for S n and S ∞. This gives: Substituting a = 8(1-r) into the first equation gives:

9. Geometric Sequences & Series ASSIGNMENT This weeks assignment is a Discussion Board activity. 5 questions have been posted in the Assignment 4 Discussion Forum and cover all aspects of Geometric Sequences and Series. As before a maximum of 3 responses to each question with everyone expected to answer one question. The assignment deadline is 5:00pm on Monday 5 April.