Calculus with Dr. Bob VI: Sequences and Series

Video Lectures

Displaying all 60 video lectures.
Lecture 1
Sequences: Definitions, Squeeze Theorem
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Sequences: Definitions, Squeeze Theorem
Calculus: We define sequences, recursive, and the notions of convergence and divergence. Techniques presented are fitting sequences to functions and the Squeeze Theorem.
Lecture 2
Examples of Sequences
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Examples of Sequences
Calculus: We define arithmetic and geometric sequences in both their function and recursive forms.
Lecture 3
Examples of Recursive Sequences
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Examples of Recursive Sequences
Calculus: Further examples of recursive sequences are presented. We also introduce the logistic sequence with various initial conditions.
Lecture 4
Sequences 1b - Squeeze Theorem/ Monotone Convergence Theorem
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Sequences 1b - Squeeze Theorem/ Monotone Convergence Theorem
Calculus: Sequences continue. An example of the Squeeze Theorem is given, we defined monotonic and bounded, and the Monotone Convergence Theorem is stated.
Lecture 5
Sequences 2 - Examples of Convergent/Monotonic/Bounded
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Sequences 2 - Examples of Convergent/Monotonic/Bounded
Calculus: For the following sequences, we consider (a) convergence/divergence, (b) monotonic increasing/decreasing, and (c) boundedness. (I) a_n = cos(n pi),
(II) a_n = cos(n pi)/n^2, (III) a_n = ln(n^{3/2})/n^{3/2}.
Lecture 6
Sequences 3 - Limit of sqrt(n^2 + n) - n
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Sequences 3 - Limit of sqrt(n^2 + n) - n
Calculus: We calculate the limit as n goes to infty of sqrt(n^2 + n) - n using two methods.
Lecture 7
Sequences 4 - Example of Monotone Convergence Theorem
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Sequences 4 - Example of Monotone Convergence Theorem
Calculus: We apply the Monotone Convergence Theorem to find the limit of the recursive sequence a_0 = 0, a_n = (1+a_{n-1})/(2+a_{n-1}).
Lecture 8
Infinite Series 1a - Definitions
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Infinite Series 1a - Definitions
Calculus: Infinite series are defined. The sum is defined as the limit of partial sums of the defining sequence on a_n. Examples given are sum (1/2)^n and sum 2^n.
Lecture 9
Infinite Series 1b - Geometric Series/ Limit Test for Divergence
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Infinite Series 1b - Geometric Series/ Limit Test for Divergence
Calculus: We continue the introduction to infinite series. A formula for the sum of a convergent geometric series is given, and the Limit Test for Divergence is presented.
Lecture 10
Infinite Series 1c - Telescoping Series
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Infinite Series 1c - Telescoping Series
Calculus: We consider telescoping series and their construction. Examples are (a) sum 1/(n^2 + n), (b) sum (2n+1)/(n^4+2n^3+n^2), and (c) sum (1/2)^{n+1}.
Lecture 11
Infinite Series 2 - Example of Convergence/Divergence
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Infinite Series 2 - Example of Convergence/Divergence
Calculus: Determine whether the following series converge or diverge: (a) sum (-3)^n, (b) sum (2^n + 3^n)/5^n, (c) sum n/(n+1), (d) sum n^2/ln(n), and (e) sum e^{-n}.
Lecture 12
Infinite Series 3 - Decimal Expansion of Fractions
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Infinite Series 3 - Decimal Expansion of Fractions
Calculus: We show how to convert repeating decimals into fractional form. Examples are .(123) and 5.23(12).
Lecture 13
Fractals
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Fractals
Calculus: We introduce elementary fractals as an application of geometric series. Examples are the Cantor set and the Sierpinski triangle.
Lecture 14
The Integral Test for Series 1a - Definition/ Examples
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The Integral Test for Series 1a - Definition/ Examples
Calculus: The Integral Test for series convergence is stated and explained. Examples are given. (a) sum 1/n, (b) sum e^{-n}.
Lecture 15
The Integral Test for Series 1b - More Examples/ p-Series
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The Integral Test for Series 1b - More Examples/ p-Series
Calculus: Further examples of the Integral Test for series convergence are presented. A special case is given as the p-series Test. (a) sum 1/(n ln(n)^2) (b) sum 1/sqrt(n), (b) sum 1/n^2, (c) sum 1/n^4.
Lecture 16
The Integral Test for Series 2 - More Examples
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The Integral Test for Series 2 - More Examples
Calculus: We determine convergence or divergence of the following examples using the Integral Test. (a) sum 1/(2n+4), (b) sum (n^2)/(n^3 + 1)^2, (c) sum 2^{-n}.
Lecture 17
Estimating Sums with the Integral Test
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Estimating Sums with the Integral Test
Calculus: We show how to use the Integral Test to estimate the error in the partial sums of a convergent series. We find n such that the nth partial sum of sum 1/n^2 is within .1 and .01 of the actual sum pi^2/6.
Lecture 18
Direct Comparison Test for Series 1
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Direct Comparison Test for Series 1
Calculus: The Direct Comparison Test for series convergence is presented and explained. Examples presented are sum 1/(n^2 + n + 1) and 1/(n^{1/3} + 6). We also show how to set up the inequalities needed for the test.
Lecture 19
Divergence of Series for 1/ln(n)
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Divergence of Series for 1/ln(n)
Calculus: The Direct Comparison Test is used to show the divergence of the series sum 1/ln(n).
Lecture 20
Limit Comparison Test for Series 1
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Limit Comparison Test for Series 1
Calculus: We present and explain the Limit Comparison Test for series convergence. The examples sum 1/(n^2 + n + 1) and sum 1/(n^{1/3} + 6) are reworked with the new test.
Lecture 21
Limit Comparison Test for Series 2
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Limit Comparison Test for Series 2
Calculus: We apply the Limit Comparison Test for series convergence to (a) sum 1/(2^n + 6) and (b) sum 1/(2^n + 3n).
Lecture 22
Rational Function Test for Series
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Rational Function Test for Series
Calculus: Let a_n = P(n)/Q(n), where P and Q are nonzero polynomials. We state a test for the convergence or divergence of the series for a_n. Examples are (a) sum (n^2 + 1)/n, (b) sum n/(n^2 + 1), and (c) sum (n^2 + 1)/(n^4 - 6).
Lecture 23
Alternating Series 1a - Alternating Series Test
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Alternating Series 1a - Alternating Series Test
Calculus: We define alternating series and give a test for convergence. We explain the test through a series of pictures. The example of sum (-1)^{n+1} 1/n is given.
Lecture 24
Alternating Series 1b - Estimating the Remainder
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Alternating Series 1b - Estimating the Remainder
Calculus: We show how to estimate the sum of a convergent alternating series within a given bound using partial sums. We apply our method to the alternating series sum (-1)^{n+1} 1/n.
Lecture 25
Alternating Series 1c - More Remainder Estimates
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Alternating Series 1c - More Remainder Estimates
Calculus: We consider two examples. The geometric series sum (-1/2)^{n+1} can also be represented as an alternating series. We apply the Alternating Series Test, and find the partial sum S_n that is within .01 of the actual sum. In the second example, we apply the AST to sum (-1)^{n+1} ln(n)/n.
Lecture 26
Absolute Convergence Test
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Absolute Convergence Test
Calculus: We show the Absolute Convergence Test for series, and define the notions of absolute and conditional convergence. We also give an example of a conditionally convergent series whose sum changes when the terms are rearranged.
Lecture 27
The Ratio Test for Series
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The Ratio Test for Series
Calculus: The Ratio Test captures convergence of a series by taking the limit of ratios of consecutive terms. A proof is given, and we consider several examples. (a) sum n/2^n, (b) sum (-1)^n 1/n!, (c) general geometric series, (d) general p-series.
Lecture 28
Series Convergence for n!/n^n
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Series Convergence for n!/n^n
Calculus: We verify the convergence of the series for n!/n^n by using the Ratio Test and the Direct Comparison Test. For the Ratio Test, we use the limit of (1+1/n)^n equals e. For the Direct Comparison Test, we compare with the p-series 2/n^2.
Lecture 29
The Root Test for Series
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The Root Test for Series
Calculus: The Root Test for convergence of series checks the limit of the nth roots of the nth term of the sequence. This series test works best when the general term has exponent n. We test the examples (a) sum 2^n/n^n, (b) sum [n/ln(n)]^n, and (c) the general geometric series.
Lecture 30
Root Test for Series Sum (1-1/n^2)^{n^3}
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Root Test for Series Sum (1-1/n^2)^{n^3}
Calculus: Determine whether the series sum (1-1/n^2)^{n^3} converges or diverges. We show convergence using the root test and an application of L'Hopital's Rule. In addition, we show two instances of substitution to evaluate limits.
Lecture 31
Series Test Round-Up 1
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Series Test Round-Up 1
Part 1: Determine if the following series converge or diverge: (1) sum 2^n(n+1)^2/n! , (2) sum 1/(sqrt(n) (1+sqrt(n))^2), (3) sum (1/2)^n - (1/2)^{n+1}, and (4) sum 1+(-1)^n.
Lecture 32
Series Test Round-Up 2
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Series Test Round-Up 2
Part 2: Determine whether the following series converge or diverge: (5) sum 1/(ln(n)^2), (6) 1 - 1/4 - 1/9 + 1/16 - 1/25 - 1/36 ..., (7) sum (n^5 + 7n + 2)/(n^3 + 3n + 4), (8) sum 3(e-2)^n.
Lecture 33
Series Test Round-Up 3
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Series Test Round-Up 3
Part 3: Determine the convergence or divergence of the series: (9) sum (-1)^n/(n+ln(n)), (10) sum 1/(n+ln(n)), (11) sum cos(n)/n^3, and (12) sum ln(n^n)/(n^{3/2} ln(n)).
Lecture 34
Motivating Taylor Polynomials 1
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Motivating Taylor Polynomials 1
We motivate the Taylor coefficient formula by showing how to recover a polynomial from its derivatives at a point.
Lecture 35
Motivating Taylor Polynomials 2
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Motivating Taylor Polynomials 2
Calculus: Continuing with Taylor polynomials, we consider a method for improving the degree of accuracy for tangent line approximations by considering higher derivatives. This leads to the definition of Taylor coefficients. we consider the examples of f(x) = x^3+1 at x=1.1 and f(x) = sin(x) at x = 3.
Lecture 36
Application of Taylor Series: Re-centering Polynomials
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Application of Taylor Series: Re-centering Polynomials
We rewrite the polynomial P(x) = 2x^3 + 2x^2 + 2x + 3 as a polynomial with center at x=1. The coefficients are calculated using the Taylor coefficient formula.
Lecture 37
Approximating with Maclaurin Polynomials
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Approximating with Maclaurin Polynomials
We calculate the 3rd Maclaurin polynomial for f(x) = ln(1-x) and use it to approximate ln(1.1). The key step is the Taylor polynomial formula.
Lecture 38
Approximating with Taylor Polynomials
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Approximating with Taylor Polynomials
We calculate the 3rd Taylor polynomial for f(x) = 1/(x+1) centered at x=1, and use it to approximate 1/(2.1).
Lecture 39
Fast Maclaurin Polynomial for Rational Function
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Fast Maclaurin Polynomial for Rational Function
Calculus: Let f(x) = (-x^2+13)/(x+2)(x-1)^2. Find the partial fraction expansion for f(x) and use it to find the second Maclaurin polynomial of f(x). We show two methods for finding the polynomial.
Lecture 40
Taylor's Theorem for Remainders
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Taylor's Theorem for Remainders
Calculus: Given a Taylor polynomial for a function f(x) with n+1 derivatives, Taylor's Theorem gives us a method for estimating the error from the actual value. The example of f(x) = x^5 + 1 is given.
Lecture 41
Taylor's Theorem : Remainder for 1/(1-x)
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Taylor's Theorem : Remainder for 1/(1-x)
Calculus: We apply Taylor's Theorem to the remainder of f(x) = 1/(1-x) with Maclaurin polynomial P_n(x) = 1 + x + x^2 + ... + x^n. We find the explicit point for equality when x = 1/2 and n = 4.
Lecture 42
Power Series 1a - Interval and Radius of Convergence
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Power Series 1a - Interval and Radius of Convergence
We define power series functions and define the interval and radius of convergence. The example of f(x) = sum x^n/n^2 is used as a concrete example for evaluating points.
Lecture 43
Power Series 1b - Interval of Convergence Using Ratio Test
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Power Series 1b - Interval of Convergence Using Ratio Test
For a given power series function, we show how to find the interval of convergence using the ratio test. Particular care must be applied to the endpoints. Examples used are (a) sum n! x^n, (b) sum (x-1)^n/n!, and (c) sum (x+2)^n/n.
Lecture 44
Example of Interval of Convergence Using Ratio Test
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Example of Interval of Convergence Using Ratio Test
Calculus: Find the interval and radius of convergence of the power series f(x) = sum (2x+1)^n/(n+1)^{1/3}. We use the ratio test to find the open interval and then check the endpoints. The radius is half the length of the interval.
Lecture 45
Power Series 1c - Interval of Convergence Using Root Test
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Power Series 1c - Interval of Convergence Using Root Test
We find the interval and radius of convergence for a power series using the root test. We consider the power series: (a) sum [ln(n)/n]^n x^n (b) sum n^n/e^2n x^n, and (c) sum (n/(n+1))^n x^n.
Lecture 46
Power Series 1d - Finding the Center
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Power Series 1d - Finding the Center
We consider power series where the (x-c) terms are not clear. Calculate the interval and radius of convergence of: (a) sum (2x + 4)/n^3, and (b) sum ((1/3) x - 2)/n^3.
Lecture 47
Power Series with Squares
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Power Series with Squares
We show how to find the interval of convergence for a power series if an x^2 term results from the ratio test. Examples are (a) sum [(x/4)^n]/n, and (b) 1 - x^2/2! + x^4/4! - ... .
Lecture 48
Derivative/Antiderivative of a Power Series 1a - Basics
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Derivative/Antiderivative of a Power Series 1a - Basics
We define the derivative and antiderivative of a power series function as a term-by -term process. The interval of convergence remains unchanged, although convergence at the endpoints may change. We revisit the example of sum (x^n)/n^2.
Lecture 49
Derivative/Antiderivative of a Power Series 1b - Interval of Convergence
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Derivative/Antiderivative of a Power Series 1b - Interval of Convergence
We show how to calculate the interval of convergence of f'(x) when the IC of f(x) is obtained by the ratio test. A similar method is used for the IC of the antiderivative of f(x). An example using the power series for f(x) = e^{x-1} is given.
Lecture 50
Derivative/Antiderivative of a Power Series 1c - More Examples
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Derivative/Antiderivative of a Power Series 1c - More Examples
We find the derivatives and antiderivatives of the power series functions (a) f(x) = 1 + x + x^2 + x^3 + ..., and (b) f(x) = 1 - x^2/2! + x^4/4! - ....
Lecture 51
Increasing the Interval of Convergence
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Increasing the Interval of Convergence
We consider the function f(x) = 1/(x+1) as represented as a power series function centered at x=0 and x=1. We note that moving the center away from the vertical asymptote allows for a bigger interval of convergence.
Lecture 52
Constructing Power Series from Functions 1a - Geometric Power Series
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Constructing Power Series from Functions 1a - Geometric Power Series
We use the geometric power series to turn rational functions into power series functions. We represent f(x) = 1/(2x+6) as power series centered at x=0 and x=1.
Lecture 53
Constructing Power Series from Functions 1b - More Geometric Power Series
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Constructing Power Series from Functions 1b - More Geometric Power Series
Continuing with constructions based on the geometric power series, we use partial fractions to find the power series of f(x) = 1/(x^2 + x) centered at x=1. As a second example, we find a series representation for pi based on a power series function for tan^{-1}(x) centered at x=0.
Lecture 54
Constructing Power Series from Functions 1c - Taylor Coefficients
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Constructing Power Series from Functions 1c - Taylor Coefficients
Calculus: We give a method for associating a power series to a function when the geometric power series does not apply. Taylor and Maclaurin series are defined, and we show how to extract coefficients from (1+x)^50 using Taylor coefficients. We also note shortcomings in assigning a series to a function.
Lecture 55
The Taylor Series for f(x) = ln(x) at x = 1
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The Taylor Series for f(x) = ln(x) at x = 1
Calculus: We derive the Taylor series for f(x) = ln(x) at x = 1 and use the 4th Taylor polynomial to estimate ln(.9). We then apply Taylor's Theorem to obtain a bound for the error.
Lecture 56
The Maclaurin Series for f(x) = 1/(1-x)^2
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The Maclaurin Series for f(x) = 1/(1-x)^2
Calculus: We find the Maclaurin series for f(x) = 1/(1-x)^2 as 1 + 2x + 3x^2 + ... by using three different methods: (a) Derivative of power series, (b) product of power series, and (c) Taylor coefficient formula.
Lecture 57
The Maclaurin Series for f(x) = e^x
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The Maclaurin Series for f(x) = e^x
Calculus: We derive the Maclaurin series for e^x and estimate e^{-1} by way of the error estimate for alternating series.
Lecture 58
The Maclaurin Series for sin(x), cos(x), and tan(x)
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The Maclaurin Series for sin(x), cos(x), and tan(x)
Calculus: We compute the Maclaurin series for f(x) = sin(x) using the Taylor coefficient formula. The series for cos(x) is obtained by differentiation. From these, we show how to divide one series into another to obtain the first few terms for the series of tan(x).
Lecture 59
The Maclaurin Series of f(x) = (1+x)^{1/2} 1a
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The Maclaurin Series of f(x) = (1+x)^{1/2} 1a
Calculus: We derive the Maclaurin series of f(x) = sqrt(1+x) and use it to compute the series for 1/sqrt(1+x). With this, we obtain the series for sin^{-1}(x), and, in turn, calculate an estimate of sin{-1}(1/2) = pi/6.
Lecture 60
The Maclaurin Series for f(x) = (1+x)^{1/2} 1b
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The Maclaurin Series for f(x) = (1+x)^{1/2} 1b
Calculus: We use the Maclaurin series for (1+x)^{1/2} to obtain an estimate for the definite integral int_0^1 (1+x^4)^{1/2} dx. A bound for the error is given using the rule for alternating series.