1. Warm up   1. Find the tenth term in the sequence:   2. Find the sum of the first 6 terms of the geometric series 2-8+32-128…   If r=-2 and a 8 = -384 what is the first term of the sequence?

2. 12.3 Infinite Sequences and Series Objective: Objective: To find the limit of the terms of an infinite sequence To find the sum of an infinite geometric series

3. Infinite Sequences & Series   Infinite sequence – a sequence that has infinitely many terms. as the “n” increases the terms decrease and get closer to 0. (no term actually becomes 0) But 0 is called the limit of the terms of the sequence.   Limits can be used to determine if a sequence approaches a value.

4. When any positive power of n appears only in the denominator of a fraction and n approaches infinity, the limit equals zero.

5. Example   Estimate the limit of the sequence:   If n=50 3.398447   If n=100 3.448374   If n=500 3.489535   If n=1000 3.494759

6. Ex 1 Estimate the limit of

7. Theorems of Limits   For sequences with more complicated general forms. Applications of the following limit theorems can make the limit easier to find.   If the exists, exists, and c is a constant, then the following theorems are true. Limit of a Sum Limit of a Difference Limit of a Product Limit of a Quotient Limit of a Constant

8. Example Find the limit

9. Example find the limit

10. Limits don’t exist for all infinite sequences. If the absolute value of a sequence becomes arbitrarily great or if the terms don’t approach a value the sequence has no limit.  Example

11. Ex 5

12.   When n is even, (-1) n = 1 and when n is odd, (-1) n = -1. Therefore the sequence would have no limit.

13. Finding the limit   If the largest exponents are the same in the numerator and the denominator, then the limit is the ratio of the coefficients of the terms containing the largest exponent.   If the largest exponent is in the numerator, then there is no limit.   If the largest exponent is in the denominator, then the limit is 0.

14. Sum of an Infinite Series   If S n is the sum of the first n terms of a series, and S is a number such that S-S n approaches zero as n increases without bound, then the sum of the infinite series is S.

15. Sum of the Terms of an Infinite Geometric Sequence The sum of the terms of an infinite geometric sequence with first term a 1 and common ratio r, where |r| < 1 is given by. Sum of an Infinite Geometric Sequence

16. Example find the sum of 60 + 24 + 9.6 + …

17. Example   Write as a fraction   So and

18. Practice   Write as a fraction.