Chapter 1 Infinite Series, Power Series January 13,15 Geometric series 1.1 The geometric series 1.2 Definitions and notation 1.3 Applications of series Infinite sequence: An ordered set of infinite number of quantities. Examples: Limit of an infinite sequence: Examples p5.1-3; Problems 2.7, 8.

Infinite series: The sum of an infinite sequence of numbers. Examples: Note: The sum of an infinite series may not be finite. Even when the sum is finite, we still cannot do it by adding the terms one by one. Partial sum of an infinite series: Sum of an infinite series is defined as:

Geometric series: A geometric series has the general term of and Partial sum of a geometric series: Sum of a geometric series: Application: Change recurring decimals into fractions. Examples:

Reading: L’Hospital’s rule:

Read: Chapter 1: 1-3 Homework: 1.1.2,8; 1.2.1,7. Due: January 24

January 17 Convergence of series 1.4 Convergent and divergent series Convergence of series: If for an infinite series we have , where S is a finite number, the series is said to be convergent. Otherwise it is divergent. Note: Whether or not a series is convergent is of essential interest for the series. For a divergent series, Sn either approaches infinity or is oscillatory. Adding or removing a finite number of terms from an infinite series will not affect whether or not it converges. Problems 4.3, 6.

1.5 Testing series for convergence; the preliminary test Preliminary test: If then the series is divergent. Note: The requirement that is a necessary condition for convergence of a series, but is not sufficient. E.g., the harmonic series. If , then further test is needed. Problems 5.3, 9.

1.6 Convergence test for series of positive terms; absolute convergence A. The comparison test: Problems 6.4, 5.

Reading: Monotone convergence theorem: Let S be a set of real numbers. A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound for S if x is an upper bound for S, and x ≤ y for every upper bound y of S. Least-upper-bound property: Any set of real numbers that has an upper bound must have a least upper bound. Monotone convergence theorem: A bounded monotonic sequence of real numbers has a finite limit.

Read: Chapter 1: 4-6 Homework: 1.4.7; 1.5.3,9; 1.6.4,5. Due: January 24

January 22, 24 Convergence test 1.6 Convergence test for series of positive terms; absolute convergence B. The integral test: Note: The lower limit in the integral is not necessary. Using x = 0 or x =1 may cause problems. E.g., Example p12; Problems 6.12.

The p-series:

C. The ratio test: Example p14.1,2; Problems 6.19,21.

D. Limit comparison test (a special comparison test): Example p15.1; Problems 6.31,35.

Read: Chapter 1:6 Homework: 1.6.12,13,21,26,28,32,35. Due: January 31

January 29 Alternating series 1.7 Alternating series 1.8 Conditionally convergent series Absolute convergence: A series is said to be absolutely convergent if is convergent. Theorem: An absolutely convergent series is convergent. Conditional convergence: A series is said to be conditionally convergent if is convergent but is not convergent.

Alternating series: An alternating series is a series whose successive terms alternate in sign. E.g., Test for alternating series (Leibnitz’s alternating series theorem): An alternating series converges if the absolute values of the terms decreases steadily to 0. That is Problems 7.4,6,7.

1.9 Useful facts about series The convergence or divergence of a series is not affected by multiplying each term by the same none-zero constant. The sum of two convergent series is convergent. Problems 9.11,18,19.

Read: Chapter 1:7-9 Homework: 1.7.3,4,6,7; 1.9.1,4,9,10,16,20. Due: February 7

1.10 Power series; interval of convergence January 31 Power series 1.10 Power series; interval of convergence Power series: A power series is an infinite series of the form We say that the power series is in x and is about the point a. A special case is a=0 and Interval of convergence: The range of x where the power series converges. Ratio test of the convergence of a power series: Example p21.2,3; Problems 10.2,7,17.

1.11 Theories about power series A power series may be differentiated or integrated term by term. The resultant series converges within the original interval of convergence (but not necessarily at the endpoints). Two power series may be added, subtracted, or multiplied. The resultant series converges at least in the common interval of convergence. Two power series can be divided if the denominator series is not 0, and the resultant series has some interval of convergence. One series can be substituted into another if the value of the substituted series is in the interval of convergence of the other series. The power series of a function is unique. These theories are easy to understand if we treat a power series as a well defined function.

1.12 Expanding functions in power series Taylor series: A Taylor series expansion of a function f (x) at x=a has the form Maclaurin series: The Taylor series expansion of a function f (x) at x=0:

1.13 Techniques for obtaining power series expansions Some important Taylor series expansions (printed in head):

Read: Chapter 1:10-13 Homework: 1.10.2,7,15,23. Due: February 7

February 3 Power series expansion 1.13 Techniques for obtaining power series expansions A. Multiply a series by a polynomial or by another series Example p26.1,2; Problem 13.5,9. B. Division of two series or of a series by a polynomial Example p27.1; Problem 13.22. C. Binomial series Example p29.2. D. Substitution of a polynomial or a series for a variable in another series Example p29.1; Problem 13.10,27. E. Combination of methods Example p30; Problem 13.13. F. Taylor series using the basic Maclaurin series Example p30.1; Problems 13.39.

1.14 Accuracy of series approximations Error in a convergent alternating series: Example p34.1; Problem 14.4. Note: This rule only applies to an alternating series.

Theorem: Problem 14.6.

Read: Chapter 1:13-14 Homework: 1.13.5,8,14,20,28,42; (Find first 3 terms. No computer work needed.) 1.14.6. Due: February 14