1. Chapter 10
Infinite Series
by: Anna Levina
edited: Rhett Chien

2. Section 10.1 Maclaurin and Taylor polynomial Approximations
Recall: Local Linear Approximation
Local Quadratic (Cubic) Approximation
Maclaurin Polynomials
Taylor Polynomials
Sigma Notation for Taylor and Maclaurin Polynomials
The nth Remainder

3. Local Linear Approximation
Local linear approximation of a function f at x0 is
f(x) = e^x
Tangent:
y = 1+x
Local linear
approximation
f(x) ≈ 1+x

4. Linear approximation works only on values close to x0.
If the graph of the function f(x) has a pronounced “bend” at x0, then we can expect that the accuracy of the local linear approximation of f at x0 will decrease rapidly as we progress away from x0.
The way to deal with this problem is to approximate the function f at x0 by a polynomial p of degree 2 with the property that the value of o and the values of its first two derivatives match those of f at x0. As a result, we can expect that the graph of p will remain closer to the graph of f over a larger interval around x0 than the graph of the local linear approximation.
Polynomial p is local quadratic approximation of f at x=x0.

5. Local Quadratic Approximation
f(x) ≈ ax^2 + bx + c
let x0 = 0
f(x0) = f(0) = 0
f’(x) = 2ax + b
f’(x0) = f’(0) = b
f”(x) = 2a
f”(x0) = f”(0) = 2a
to find a, b, c:
f(0) = c
f’(0) = b
f”(0)/2 = a

6. Visualization
y = e^x
linear: y = 1 + x
quadratic:

7. Maclaurin Polynomials
The accuracy of the approximation increases as the degree of the polynomial increases.
We use Maclaurin polynomial.
If f can be differentiated n times at 0, then we define the nth Maclaurin polynomial to be

8. Taylor Polynomials
If f can be differentiated n times at a, then we define the
nth Tylor polynomial for f about x = a to be

9. Example Find the Maclaurin polynomial of order 2 for e^(3x)
f(0) = 1 = c
f’(0) = 3 = b
f”(0) = 9 = 2a
p2(x) = 1 + 3x + 9(x^2)/2

10. Example Find a Taylor polynomial for f(x) = 3lnx of order 2 about x=2
f(2) = 3ln2
f’(2) = 3/2
f” (2) = - 3/4
p2 = 3ln2 + 3/2(x-2) – 3/8(x-2)^2

11. Sigma Notation for Taylor and Maclaurin Polynomials
We can write the nth-order Maclaurin polynomial for f(x) as
We can write the nth-order Taylor polynomial for f(x) about c as

12. The nth Remainder
If the function f can be differentiated n+1 times on an interval I containing the number x0, and if M is an upper bound for
on I,
≤ M for all x in I, then
for all x in I.

13. Just a cool picture

14. Section 10.2 Sequences Definition of a sequence Limit of a sequence
The squeezing theorem for sequences

15. Definition of a Sequence
A sequence is a function whose domain is a
set of integers. Specifically, we will regard
the expression {an}n=1 to be an alternative
notation for the function
f(n) = an, n=1,2,3,…

16. Limit of a sequence
A sequence {an} is said to converge to the limit L if given any ε>0, there is a positive integer N such that Ian – LI < ε for n≥N. In this case we write
A sequence that does not converge to some finite
limit is said to diverge

17. The squeezing theorem for sequences
Let {an}, {bn}, and {cn}, be sequences such that an≤ bn≤cn
If sequences {an} and {cn} have a common limit L as n→ +∞, then {bn} also has the limit L as n→+∞.

18. Example
The general term for the
sequence 3, 3/8, 1/9, 3/64,… is
3/n^3

19. Example
Show that +∞
{ln(n)/n} converges.
n=1
What is the limit?

20. Section 10.3 Monotonic sequences
Strictly monotonic
Monotonic

21. Definition A sequence {an}n=1 is called
Strictly increasing if a1 < a2 < a3 < … < an< …
Increasing if a1≤ a2≤ a3 ≤ … an ≤ …
Strictly decreasing if a1 > a2 > a3 > … an > …
Decreasing if a1≥ a2 ≥ a3 ≥ … an ≥ …

22. Testing for Monotonicity
Method 3 (Ratio)
an+1/an< 1
Method 1.
By inspection
Method 2
an+1 > an
Method 4
an = 1/n
Let f(x)=1/x
f’(x)= -x^(-2)
f’(x) < 0 for all x≥ 1
Therefore an is strictly decreasing

23. Eventually
If discarding infinitely many terms from a beginning of a sequence produces a sequence with certain property, then the original sequence is said to have that property eventually.

24. Example Determine which answer best describes the sequenced +∞ {6/n}
B. Strictly decreasing
A. Strictly increasing
D. Decreasing
C. Increasing

25. Section 10.4 Infinite Series
Sums of infinite series
Geometric series
Telescoping sums
Harmonic series

26. Sum = a0 + a1 + a2 +…. an Partial sum
Sn = a1 + a an,    the nth partial sum.

27. Convergent series
If Sn exists, we say that an is a convergent series, and write
Sn = an.
Thus a series is convergent if and only if it's sequence of partial sums is convergent. The limit of the sequence of partial sums is the sum of the series. A series which is not convergent, is a divergent series.

28. Geometric Series

29. Telescoping sums
A sum in which subsequent terms cancel each other, leaving only initial and final terms. For example,
S= = =
is a telescoping sum.

30. Harmonic Series
Always diverge

31. Example
The sum
is convergent with sum 1. =
Sn = -
= 1 -
1    as    n

32. Section 10.5 Convergence tests
Divergence test
Integral test
P-series
The comparison test
The limit comparison test
The ratio test
Root test

33. Divergence test If the series converges, then the
sequence converges to zero. Equivalently:
If the sequence does not converge to
zero, then the series
can not converge.

34. Integral test converges if and only if converges
Suppose that f(x) is positive, continuous, decreasing function on the interval [N, ). Let a n = f(n). Then
       converges if and only if           converges

35. p-Series The series is called a p-Series.
if p > 1 the p-series converges
if p ≤ 1 the p-series diverges

36. Comparison test
Suppose that converges absolutely, and is a sequence of numbers for which
| bn |
| an | for all n > N
Then the series
converges absolutely as well.
If the series
converges to positive infinity, and
is a sequence of numbers for which
for all n > N
Then the series
also diverges.

37. Limit Comparison Test Suppose and are two infinite
series. Suppose also that
r = lim | a n / b n | exists, and 0 < r <
Then converges absolutely if and only
if converges absolutely.

38. Ratio test Consider the series . Then
if lim | a n+1 / a n | < 1 then the series converges absolutely. n
if there exists an N such that | a n+1 / a n |   1 for all n > N then the series diverges.
if lim | a n+1 / a n | = 1, this test gives no information

39. Root test Consider the series . Then:
if lim sup | a n |^ (1/n) < 1 then the series converges absolutely.
if lim sup | a n |^ (1/n) > 1 then the series diverges
if lim sup | a n |^ (1/n) = 1, this test gives no information

40. Example
Does Euler's series        converge ?
The series
is called Euler's series. It converges to Euler's number e.

41. Example Does the series converge or diverge ?
We will use the limit comparison test, together with the p-series test.
First, note that 1 / (1 + n^ 2) < 1 / n^ 2
But since the series
  / n 2 is a p-series with p = 2, and therefore converges,
the original series
must also converge by the comparision test.

42. Example
Determine if the following series is convergent or divergent

43.                                               .
Since
we conclude, from the Ratio-Test, that the series
is convergent.

44. Example
Determine whether series converges and find the sum.

45. Section Next Alternating Series. Conditional Convergence
AST
Absolute Convergence
Conditional Convergence
The Ratio Test for absolute convergence

46. Alternating Series
A series of a form or
Where

47. Alternating Series Test
Also known as the Leibniz criterion. An alternating series converges if
and
                

48. Absolute convergence
A series is said to converge absolutely if the series converges, where denotes the absolute value.
If a series is absolutely convergent, then the sum is independent of the order in which terms are summed.
Furthermore, if the series is multiplied by another absolutely convergent series, the product series will also converge absolutely.

49. Conditional Convergence
If the series converges, but
does not, where is the absolute value, then the series is said to be conditionally convergent.

50. The ration test for absolute convergence
The same as ratio test, just use absolute value.
If the series diverges absolutely, check for conditional convergence using another method.

51. Example Classify the series as either absolutely
convergent, conditionally convergent, or
divergent.
by the Alternating Series Test, the series
is convergent. Note that it is not absolutely convergent.

52. Section 10.8 Maclaurin and Taylor Series; Power series
Radius and interval of convergence
Function defined by power series

53. Taylor Series
If f has derivatives of all orders xo, then we call the series
the Taylor Series for f about x=x0.

54. Maclaurin Series A Maclaurin series is a Taylor series
of a function f about 0
                                                                                                    

55. Example Find the Taylor series with center x=x0 for so f(0)=1.

56. Power Series
A power series about a, or just power series, is any series that
can be written in the form, where a and cn are numbers.

57. Radius and interval of convergence
For any power series in x, exactly one of the
following is true:
The series converges only for x=0.
The series converges absolutely (and hence converges) for all real values of x.
The series converges absolutely (and hence converges) for all x in some finite open interval (-R,R). At either of the values x=R or x=-R, the series may converge absolutely, converge conditionally, or diverge, depending on the particular series.

58. Finding the interval of convergence
Use ratio test for absolute convergence
If p = then the series
is convergent.
Find values of IxI for which p<1

59. Function defined by power series
If a function f is expressed as a power series on some interval, then we say that f is represented by the power series on the interval.

60. Some series to remember
Dollars equal cents
Theorem: 1$ = 1c. Proof: And another that gives you
a sense of money disappearing. 1$ = 100c = (10c)^2 = (0.1$)^2 = 0.01$ = 1c = = = = =

61. Example Find the radius of convergence
The general term of the series has
the form
Consequently, the radius of convergence equals 1

62. Measuring infinity

63. Section 10.9 Convergence of Taylor Series
The nth remainder
Estimating the nth remainder
Approximating different functions

64. The nth Remainder Problem.
Given a function f that has derivatives of all orders at x = x0, determine whether there is an open interval containing x0 such that f(x) is the sum of its Taylor series about x=x0 at each number in the interval; that is
for all values of x in the interval.

65.                                                                                                                       
Lagrange Remainder

66. Section 10.10 differentiating and integrating power series
Differentiating power series
Integrating power series

67. Differentiation

68. Integration

69. Examples online

70. Bibliography http://mathworld.wolfram.com/
Anton, Bivens, Davis. Calculus. Anton textbooks, 2002.