1. 12.3: Infinite Sequences and Series

2. Consider the following sequence
1, 1 2 , 1 3 , 1 4 , 1 5 ,…
Each term of this sequence is of the form 1 𝑛

3. What happens to these terms as n gets very large?
In general, the lim 𝑛→∞ 1 𝑛 𝑟 =0 , for all positive r

4. Many sequences have limiting factors
7, 7 4 , , , …

5. Estimate the limit for the following sequence
9 5 , 16 4 , ,… 7 𝑛 𝑛 2 +3𝑛

6. Find the limit
lim 𝑛→∞ 𝑛 2 𝑛 2

7. Find the limit
lim 𝑛→∞ 5 𝑛 2 +𝑛−4 𝑛 2 +1

8. Find the limit
lim 𝑛→∞ 4 𝑛 2 +5𝑛+2 2𝑛

9. Find the limit
lim 𝑛→∞ 𝑛 2 2𝑛

10. Find the limit
lim 𝑛→∞ 2𝑛 𝑛 2

11. Find the limit
lim 𝑛→∞ 𝑛(−1) 𝑛 8𝑛+1

12. In general;
If the highest exponent of the numerator and denominator are the same, the limit is the ratio of their coefficients
If the exponent of the numerator is higher, the limit DNE
If the exponent of the denominator is higher, the limit is zero

13. 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 approaches infinity, then the sum of the infinite series is S.
lim 𝑛→∞ 𝑆 𝑛 =𝑆

14. For very large n values…
If lim 𝑛→∞ 𝑎 𝑛 ≠0 , the series has no sum.
If lim 𝑛→∞ 𝑎 𝑛 =0 , the series MAY have a sum.

15. Recall the geometric series
7, 7 4 , , , …
What happens to the terms as n increases?

16. Sum of an infinite geometric series
The sum, S, of an infinite geometric series where 𝑟 <1, is given by S = 𝑎 1 1−𝑟

17. Find the infinite sum of the series below…

18. Find the infinite sum of the series below…

19. Find the infinite sum of the series below…

20. Can 0. 762 be written as a fraction?

21. Homework:
Section 12.3, pages 781: 14-19, 24, 25,