1. (a) an ordered list of objects.
12. 1 A sequence is…
(a) an ordered list of objects.
(b) A function whose domain is a set of integers.
Domain: 1, 2, 3, 4, …,n…
Range a1, a2, a3, a4, … an…
{(1, 1), (2, ½), (3, ¼), (4, 1/8) ….}

2. Describe a pattern for each sequence. Write a formula for the nth term
Finding patterns
Describe a pattern for each sequence. Write a formula for the nth term

3. Write the first 5 terms for
On a number line
As a function
The terms in this sequence get closer and closer to 1. The sequence CONVERGES to 1.

4. The terms in this sequence do not get close to
Write the first 5 terms
The terms in this sequence do not get close to
Any single value. The sequence Diverges

5. The sequence converges to 3.
Write the terms for an = 3
The terms are 3, 3, 3, …3
The sequence converges to 3.

6. y= L is a horizontal asymptote when sequence converges to L.

7. A sequence that diverges

8. Write the first 5 terms of the sequence.
Sequences
Write the first 5 terms of the sequence.
Does the sequence converge? If so, find the value.
The sequence converges to 0.
The sequence diverges.

9. Represents the sum of the terms in a sequence.
12.2 Infinite Series
Represents the sum of the terms in a sequence.
We want to know if the series converges to
a single value i.e. there is a finite sum.
The series diverges because sn = n. Note that the
Sequence {1} converges.

11. If the sequence of partial sums converges, the series converges
Partial sums of
and
If the sequence of partial sums converges, the series converges
Converges to 1 so series converges.

12. Can use partial fractions to rewrite
Finding sums
Can use partial fractions to rewrite

13. The partial sums of the series.
Limit

14. Find r (the common ratio)
Geometric Series
Each term is obtained from the preceding number by multiplying by the same number r.
Find r (the common ratio)

15. Write using series notation
Is a Geometric Series
Where a = first term and r=common ratio
Write using series notation

16. The sum of a geometric series
Sum of n terms
Multiply each term by r
subtract
Geometric series converges to
If r>1 the geometric series diverges.

17. Find the sum of a Geometric Series
Where a = first term and r=common ratio
The series diverges.

18. Repeating decimals-Geometric Series
The repeating decimal is equivalent to 8/99.

19. Series known to converge or diverge
1. A geometric series with | r | <1 converges
2. A repeating decimal converges
3. Telescoping series converge
A necessary condition for convergence:
Limit as n goes to infinity for nth term in sequence is 0.
nth term test for divergence:
If the limit as n goes to infinity for the nth term is not 0, the series DIVERGES!

20. Convergence or Divergence?

21. A monotonic sequence that is bounded Is convergent.
A sequence in which each term is less than or equal to the one before it is called a monotonic non-increasing sequence. If each term is greater than or equal to the one before it, it is called monotonic non-decreasing.
A monotonic sequence that is bounded
Is convergent.
A series of non-negative terms converges
If its partial sums are bounded from above.

22. 12. 3 The Integral Test
Let {an} be a sequence of positive terms. Suppose that an = f(n) where f is a continuous positive, decreasing function of x for all xN. Then the series and the corresponding integral shown both converge of both diverge.
f(n)
f(x)

23. The series and the integral both converge or both diverge
Area in rectangle corresponds to term in sequence
Exact area under curve is between
If area under curve is finite, so is area in rectangles
If area under curve is infinite, so is area in rectangles

24. Using the Integral test
The improper integral diverges
Thus the series diverges

25. Using the Integral test
The improper integral converges
Thus the series converges

26. Harmonic series and p-series
Is called a p-series
A p-series converges if p > 1 and diverges
If p < 1 or p = 1.
Is called the harmonic series and it diverges since p =1.

27. Identify which series converge and which diverge.

28. 12. 4Direct Comparison test
be a series with no negative terms
Let
Converges if there is a series
Where the terms of an are less than or equal to the terms of cn for all n>N.
Diverges if there is a series
Where the terms of an are greater than or equal to the terms of dn for all n>N.

29. Limit Comparison test and and Amd and Diverges. Limit Comparison test
Then the following series
both converge or both diverge:
and
Converges then
Converges.
Amd
and
Diverges.
Diverges then

30. Convergence or divergence?

31. A series in which terms alternate in sign
Alternating Series
A series in which terms alternate in sign or

32. Alternating Series Test
Converges if:
an is always positive
an an+1 for all n  N for some integer N.
an0
If any one of the conditions is not met, the
Series diverges.

33. Absolute and Conditional Convergence
A series is absolutely convergent if the corresponding series of absolute values converges.
A series that converges but does not converge absolutely, converges conditionally.
Every absolutely convergent series converges. (Converse is false!!!)

34. Is the given series convergent or divergent
Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent?

35. This is not an alternating series, but since
a) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent?
This is not an alternating series, but since
Is a convergent geometric series, then the given
Series is absolutely convergent.

36. Converges by the Alternating series test.
b) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent?
Converges by the Alternating series test.
Diverges with direct comparison with the harmonic
Series. The given series is conditionally convergent.

37. By the nth term test for divergence, the series Diverges.
c) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent?
By the nth term test for divergence, the series
Diverges.

38. Converges by the alternating series test.
d) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent?
Converges by the alternating series test.
Diverges since it is a p-series with p <1. The
Given series is conditionally convergent.

39. Let be a series with positive terms and
The Ratio Test
Let be a series with positive terms and
Then
The series converges if ρ < 1
The series diverges if ρ > 1
The test is inconclusive if ρ = 1.

40. Let be a series with non-zero terms and
The Root Test
Let be a series with non-zero terms and
Then
The series converges if L< 1
The series diverges if L > 1 or is infinite
The test is inconclusive if L= 1.

41. Convergence or divergence?

42. Procedure for determining Convergence.
Procedure for determining Convergence

43. Power Series (infinite polynomial in x)
Is a power series centered at x = 0.
and
Is a power series centered at x = a.

44. Examples of Power Series
Is a power series centered at x = 0.
and
Is a power series centered at x = -1.

45. Geometric Power Series

46. The graph of f(x) = 1/(1-x) and four of its polynomial approximations.
The graph of f(x) = 1/(1-x) and four of its polynomial approximations

47. Convergence of a Power Series
There are three possibilities
1)There is a positive number R such that the series diverges for |x-a|>R but converges for |x-a|<R. The series may or may not converge at the endpoints, x = a - R and x = a + R.
2)The series converges for every x. (R = .)
3)The series converges at x = a and diverges elsewhere. (R = 0)

48. What is the interval of convergence?
Since r = x, the series converges |x| <1, or
-1 < x < 1. In interval notation (-1,1).
Test endpoints of –1 and 1.
Series diverges
Series diverges

49. Geometric Power Series
Find the function
Find the radius of convergence

50. Geometric Power Series
Find the radius of convergence
Find the interval of convergence
For x = -2,
Geometric series with r < 1, converges
For x = 4
By nth term test, the series diverges.
Interval of convergence

51. Finding interval of convergence
Use the ratio test:
R=1
(-1, 1)
For x = 1
For x = -1
Interval of convergence
[-1, 1)
Harmonic series
diverges
Alternating Harmonic series converges

52. Differentiation and Integration of Power Series
If the function is given by the series
Has a radius of convergence R>0, the on the interval (c-R, c+R) the function is continuous,
Differentiable and integrable where:
The radius of convergence is the same but the interval of convergence may differ at the endpoints.

53. Constructing Power Series
If a power series exists has a radius of convergence = R
It can be differentiated
So the nth derivative is

54. Finding the coefficients for a Power Series
All derivatives for f(x) must equal the series
Derivatives at x = a.

55. If f has a series representation centered at x=a, the series must be
If f has a series representation centered at x=0, the series must be

56. Form a Taylor Polynomial of order 3 for sin x at a =
f(n)(x)
f(n)(a)
f(n)(a)/n!
sin x 1
cos x 2
-sin x 3
-cos x

57. The graph of f(x) = ex and its Taylor polynomials

58. Find the derivative and the integral

59. Taylor polynomials for f(x) = cos (x)

60. Converges only at x = 0