1. The sum of the infinite and finite geometric sequence

2. The sum of the first n terms of a sequence is represented by summation notation.
upper limit of summation
lower limit of summation
index of summation 2

3. The sum of a finite geometric sequence is given by
= ?
n = 8
a1 = 5

4. If |r| < 1, then the infinite geometric series
The sum of the terms of an infinite geometric sequence is called a geometric series.
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r a1rn
has the sum

5. Example: Find the sum of
The sum of the series is

6. Convergent and Divergent Series

7. Convergent and Divergent Series
If the infinite series has a sum, or limit, the series is convergent.
If the series is not convergent, it is divergent.

8. Ways To Determine Convergence/Divergence
1. Arithmetic – since no sum exists, it diverges
2. Geometric:
If |r| > 1, diverges
If |r| < 1, converges since the sum exists
3. Ratio Test (discussed in a few minutes)

9. Example
Determine whether each arithmetic or geometric series is convergent or divergent.
1/8 + 3/20 + 9/ /
r=6/5  |r|>1  divergent
Arithmetic series  divergent
/5 + 13/
r=1/5  |r|<1  convergent

10. Other Series
When a series is neither arithmetic or geometric, it is more difficult to determine whether the series is convergent or divergent.

11. Ratio Test
In the ratio test, we will use a ratio of an and an+1 to determine the convergence or divergence of a series.
Leading coefficient is a
Review:
Leading coefficient is d
Denominator degree is greater

12. Test for convergence or divergence of:
Since this ratio is less than 1, the series converges.

13. The ratio of the leading coefficients is 1
Test for convergence or divergence of:
The ratio of the leading coefficients is 1
Since this ratio is less than 1, the series converges.

14. Test for convergence or divergence of:
Coefficient of n2 is 1
Since this ratio is 1, the test is inconclusive.
Coefficient of n2 is 1

15. Example
Use the ratio test to determine if the series is convergent or divergent.
1/2 + 2/4 + 3/8 + 4/
Since r<1, the series is convergent.

16. Example
Use the ratio test to determine if the series is convergent or divergent.
1/2 + 2/3 + 3/4 + 4/
Since r=1, the ratio test provides no information.

17. Example
Use the ratio test to determine if the series is convergent or divergent.
2 + 3/2 + 4/3 + 5/
Since r=1, the ratio test provides no information.

18. Example
Use the ratio test to determine if the series is convergent or divergent.
3/4 + 4/16 + 5/64 + 6/
Since r<1, the series is convergent.

19. Example
Use the ratio test to determine if the series is convergent or divergent.
Since r<1, the series is convergent.

20. Example
Use the ratio test to determine if the series is convergent or divergent.
Since r>1, the series is divergent.