1. Math 181
11.2 –Series

2. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛

3. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛
series

4. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛
series
sum

5. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛
series
sum

6. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛
series
sum

7. A __________ is the ______ of the terms of a sequence
A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛
series
sum

8. What does the following series add up to
What does the following series add up to? …+ 1 2 𝑛 +… 𝑛=1 ∞ 1 2 𝑛 Let’s try adding the terms one at a time:

9. What we just calculated are called the _____________ of the series
What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

10. partial sums
What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

11. partial sums
What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

12. partial sums
What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

13. The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ .

14. The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ .
converges

15. The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ .
converges
diverges

16. We can visualize the partial sums on a coordinate system, with 𝑛 on the 𝑥-axis, and 𝑠 𝑛 on the 𝑦-axis.

17. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.

18. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series

19. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series

20. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series

21. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series
common ratio

22. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series
common ratio

23. Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
… is an example of a _______________, which generally has the form:
𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +…
𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏
Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.
geometric series
common ratio

24. When does a geometric series 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏 converge or diverge?

25. 𝑛=1 ∞ 𝑎 𝑟 𝑛−1 =𝑎+𝑎𝑟+𝑎 𝑟 2 +…
converges to 𝑎 1−𝑟 if 𝑟 <1
diverges if 𝑟 ≥1

26. Ex 1. Find the sum of 5− − …

27. Ex 1. Find the sum of 5− − …

28. Ex 2. Express 2.3 17 =2.317171717… as the ratio of two integers.

29. “Telescoping Sums”
Ex 3. Find the sum of 𝑛=1 ∞ 1 𝑛 𝑛+1 .

30. Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0
Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0. ex: 𝑛=1 ∞ 1 𝑛 𝑛+1 converges, so lim 𝑛→∞ 1 𝑛 𝑛+1 =0.

31. Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0
Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0. ex: 𝑛=1 ∞ 1 𝑛 𝑛+1 converges, so lim 𝑛→∞ 1 𝑛 𝑛+1 =0.

32. Test for Divergence: If lim 𝑛→∞ 𝑎 𝑛 does not exist or if lim 𝑛→∞ 𝑎 𝑛 ≠0, then 𝑛=1 ∞ 𝑎 𝑛 diverges. Ex 4. Does 𝑛=1 ∞ 𝑛 2 converge or diverge? Ex 5. Does 𝑛=1 ∞ 𝑛 2 5 𝑛 2 +4 converge or diverge? Ex 6. Does 𝑛=1 ∞ −1 𝑛+1 converge or diverge?

33. Test for Divergence: If lim 𝑛→∞ 𝑎 𝑛 does not exist or if lim 𝑛→∞ 𝑎 𝑛 ≠0, then 𝑛=1 ∞ 𝑎 𝑛 diverges. Ex 4. Does 𝑛=1 ∞ 𝑛 2 converge or diverge? Ex 5. Does 𝑛=1 ∞ 𝑛 2 5 𝑛 2 +4 converge or diverge? Ex 6. Does 𝑛=1 ∞ −1 𝑛+1 converge or diverge?

34. Note: Caution. Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges
Note: Caution! Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges! ex: Let’s show that even though lim 𝑛→∞ 1 𝑛 =0, 𝑛=1 ∞ 1 𝑛 diverges:

35. Note: Caution. Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges
Note: Caution! Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges! ex: Let’s show that even though lim 𝑛→∞ 1 𝑛 =0, 𝑛=1 ∞ 1 𝑛 diverges:

36. Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1
Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1. ( 𝑎 𝑛 ± 𝑏 𝑛 ) = 𝑎 𝑛 ± 𝑏 𝑛 2. 𝑘 𝑎 𝑛 =𝑘 𝑎 𝑛 (𝑘 is a constant)

37. Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1
Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1. ( 𝑎 𝑛 ± 𝑏 𝑛 ) = 𝑎 𝑛 ± 𝑏 𝑛 2. 𝑘 𝑎 𝑛 =𝑘 𝑎 𝑛 (𝑘 is a constant)

38. Ex 7. Does 𝑛=1 ∞ 3 𝑛 + 2 𝑛 5 𝑛 converge or diverge
Ex 7. Does 𝑛=1 ∞ 3 𝑛 + 2 𝑛 5 𝑛 converge or diverge? If it converges, find its sum.

39. Ex 8. Does 𝑛=1 ∞ 4 2 𝑛 converge or diverge
Ex 8. Does 𝑛=1 ∞ 𝑛 converge or diverge? If it converges, find its sum.