Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin

Chapter 10: Infinite Sequences and Series Sequences Infinite Series The Integral Test Comparison Tests The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series The Binomial Series and Applications of Taylor Series

Chapter 10 Overview Infinite sequences of numbers/(variable terms) can be used to define important constants/functions. Infinite sequences of polynomial terms are used as replacements for major transcendental functions. Because these replacements are polynomials, calculus is easy to apply. Truncated infinite sequences of variable terms are used by calculating machines to compute approximate values of transcendental functions. Some infinite sequences of numbers converge to finite values. Much of this chapter is devoted to determining which infinite sequences have this property and finding the convergence value.

10.01: Sequences 1 An Infinite Sequence of numbers is a function whose domain is the set of natural numbers {1, 2, 3, …} and whose range is provided by either a recursive or explicit algebraic rule. TI-84 sequence graphing. Definition: The Infinite Sequence {a n } converges to the number L if for every positive number ε there is an integer N such that when n > N → │ a n - L │ < ε. Proving that. If an Infinite Sequence does not converge then it diverges. Examples 1 & 2

10.01: Sequences 2 A set of theorems defining the algebra of limits allows many new limits to be computed from previously known limits (Theorem 1, section 02.02). Example 3 –Sum/Difference Rule –Product Rule –Quotient Rule –Constant Multiple Rule The Sandwich Theorem (Theorem 4, section 02.02) Example 4 Composition Rule (Theorem 10, section 02.05) Examples 5 & 6 The connection between the limit of a sequence and the limit of a related continuous function. L’Hôpital’s Rule (Theorem 6, section 04.05) Examples 7 & 8 Commonly occurring limits. Example 9

10.01: Sequences 3 Recursive definitions for infinite sequences of numbers have two parts. Example The algebraic rule. 2. The initial values needed to begin the recursion. Bounded sequences – (Upper Bounds)/(Least Upper Bounds) and (Lower Bounds)/(Greatest Lower Bounds). Example 11 Monotonic sequences – Nondecreasing and Nonincreasing sequences. Example 12 The Bounded and Monotonic Sequence theorem.

10.02: Infinite Series 1 The n th Partial Sum (s n ) of an Infinite Sequence (A) is the sum of the first n terms of that Infinite Sequence. All the n th partial sums of an Infinite Sequence form a new Infinite Sequence (P). An Infinite Series (S) is the sum of an Infinite Sequence (A). S is not a sequence. Summation notation is used to mathematically write the form of a particular Infinite Series. Definition: An Infinite Series is said to converge if the corresponding Infinite Sequence of Partial Sums (P) converges.

10.02: Infinite Series 2 Original Sequence: A = a 1, a 2, a 3, … Corresponding Series: S = a 1 + a 2 + a 3 + … Partial Sum: s n = a 1 + a 2 + a 3 + … + a n Partial Sum Sequence: P = s 1, s 2, s 3, …, s n

10.02: Infinite Series 3 Arithmetic Sequence/Series formulas. Geometric Sequence/Series formulas. Examples 1 – 4 Telescoping Series. Example 5 Series convergence implies that the corresponding Infinite Sequence (A) converges to zero (Theorem 7). Thus if the Infinite Sequence (A) does not have a limit of zero the corresponding series cannot converge (The n th - Term Test for Divergence). The converse of Theorem 7 is not true. Examples 6 – 8

10.02: Infinite Series 4 Two convergent series may be combined as a sum/difference and the new limit will be the sum/difference of the original limits. A convergent series may be multiplied by a constant and the new limit will be the original limit times the constant (Theorem 8). Example 9 A finite number of terms may be added to or deleted from a convergent sequence without changing the fact of its convergence but the convergence value may change. Reindexing a series can be used produce a simpler algebraic rule and will not change its convergence value. Example 10

10.03: The Integral Test 1 Series with Non-Negative terms → Partial Sums will be Non-Decreasing. If these Partial Sums are bounded above then the original series must converge. Example 1 Series convergence is linked to the convergence of the a Type I Improper Integral by the Integral Test (Theorem 9). The Integral Test does not yield a convergence value. Examples 2 & 4 The p-Series and Harmonic Series (Example 3, section 08.07) Example 3

10.04: Comparison Tests 1 Series with Non-Negative terms → Partial Sums will be Non-Decreasing. The Comparison Test (Theorem 10). The Comparison Test does not yield a convergence value. Like the Sandwich Theorem the difficulty is in finding the ‘bread’. Example 1 The Limit Comparison Test (Theorem 11). The Limit Comparison Test does not yield a convergence value. Examples 2 & 3

10.05: The Ratio and Root Tests 1 The Ratio Test (Theorem 12). The Ratio Test does not yield a convergence value. Example 1 The Root Test (Theorem 13). The Root Test does not yield a convergence value. Examples 2 & 3

10.06: Alternating Series, Absolute and Conditional Convergence 1 Definition: A series of alternating positive and negative terms is called an Alternating Series. The Alternating Series Test (Leibniz’s Test). Three conditions must be shown to be true to use this test. Examples 1 & 2 Definition: A series converges absolutely if the corresponding series of absolute values converges. Definition: A series that converges but does not converge absolutely converges conditionally. The Absolute Convergence Test: If the series of absolute values of a series converges then the series also converges. Examples 4 & 5

10.06: Alternating Series, Absolute and Conditional Convergence 2 Rearrangement of Absolutely Convergent Series: If a series converges absolutely then any rearrangement of the series converges absolutely to the original convergence value. Rearrangement of Conditionally Convergent Series: Rearrangements can be made to yield any finite value. Summary Table of Convergence Tests.

10.07: Power Series 1 An infinite series of the form is called a power series centered about point ‘a’. Power series are used extensively in Calculus as replacements for difficult and transcendental functions. Two problems present themselves when dealing with power series. The first problem is finding the appropriate power series. The second problem is finding the radius of convergence or range of values for which the series yields the same values as the original function.

10.07: Power Series 2 Testing a Power Series for convergence. Examples 1 – 3 There are three convergence possibilities for a Power Series (Corollary to Theorem 18). Multiplication for Power Series. Term-by-term differentiation of Power Series. Example 4 Term-by-term integration of Power Series. Examples 5 & 6 The Differentiation and Integration theorems above apply only to Power Series.

10.08: Taylor and Maclaurin Series 1 A function that has derivatives of all orders can be expressed as power series by using Taylor’s Theorem. Example 1 The Taylor series generated by f at x = a: Finite portions of Taylor series are called Taylor polynomials. Examples 2 & 3

10.09: Convergence of Taylor Series 1 Using known series (table in back of text) to find new Taylor series by Addition, Subtraction, Multiplication, and Substitution. Example 4, #2 – page 613

10.10: The Binomial Series and Applications of Taylor Series 1 Deriving the Binomial Series with Taylor’s Formula. Examples 1 & 2 The relationship between the Binomial Series and the binomial expansion of (1+x) m when m is a nonnegative integer. Evaluating nonelementary integrals. Examples 3 & 4 Evaluating Indeterminate Forms using power series. Examples 5 – 7

10.10: The Binomial Series and Applications of Taylor Series 2 Euler’s Identity: The most beautiful equation: Table of frequently used Taylor series.