  # Calculus with Dr. Bob VI: Sequences and Series

## Video Lectures

Displaying all 60 video lectures.
 Lecture 1 Play Video Sequences: Definitions, Squeeze TheoremCalculus: We define sequences, recursive, and the notions of convergence and divergence. Techniques presented are fitting sequences to functions and the Squeeze Theorem. Lecture 2 Play Video Examples of SequencesCalculus: We define arithmetic and geometric sequences in both their function and recursive forms. Lecture 3 Play Video Examples of Recursive SequencesCalculus: Further examples of recursive sequences are presented. We also introduce the logistic sequence with various initial conditions. Lecture 4 Play Video Sequences 1b - Squeeze Theorem/ Monotone Convergence TheoremCalculus: Sequences continue. An example of the Squeeze Theorem is given, we defined monotonic and bounded, and the Monotone Convergence Theorem is stated. Lecture 5 Play Video Sequences 2 - Examples of Convergent/Monotonic/BoundedCalculus: 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 Play Video Sequences 3 - Limit of sqrt(n^2 + n) - nCalculus: We calculate the limit as n goes to infty of sqrt(n^2 + n) - n using two methods. Lecture 7 Play Video Sequences 4 - Example of Monotone Convergence TheoremCalculus: 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 Play Video Infinite Series 1a - DefinitionsCalculus: 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 Play Video Infinite Series 1b - Geometric Series/ Limit Test for DivergenceCalculus: 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 Play Video Infinite Series 1c - Telescoping SeriesCalculus: 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 Play Video Infinite Series 2 - Example of Convergence/DivergenceCalculus: 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 Play Video Infinite Series 3 - Decimal Expansion of FractionsCalculus: We show how to convert repeating decimals into fractional form. Examples are .(123) and 5.23(12). Lecture 13 Play Video FractalsCalculus: We introduce elementary fractals as an application of geometric series. Examples are the Cantor set and the Sierpinski triangle. Lecture 14 Play Video The Integral Test for Series 1a - Definition/ ExamplesCalculus: The Integral Test for series convergence is stated and explained. Examples are given. (a) sum 1/n, (b) sum e^{-n}. Lecture 15 Play Video The Integral Test for Series 1b - More Examples/ p-SeriesCalculus: 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 Play Video The Integral Test for Series 2 - More ExamplesCalculus: 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 Play Video Estimating Sums with the Integral TestCalculus: 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 Play Video Direct Comparison Test for Series 1Calculus: 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 Play Video 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 Play Video Limit Comparison Test for Series 1Calculus: 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 Play Video Limit Comparison Test for Series 2Calculus: 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 Play Video Rational Function Test for SeriesCalculus: 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 Play Video Alternating Series 1a - Alternating Series TestCalculus: 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 Play Video Alternating Series 1b - Estimating the RemainderCalculus: 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 Play Video Alternating Series 1c - More Remainder EstimatesCalculus: 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 Play Video Absolute Convergence TestCalculus: 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 Play Video The Ratio Test for SeriesCalculus: 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 Play Video Series Convergence for n!/n^nCalculus: 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 Play Video The Root Test for SeriesCalculus: 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 Play Video 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 Play Video Series Test Round-Up 1Part 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 Play Video Series Test Round-Up 2Part 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 Play Video Series Test Round-Up 3Part 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 Play Video Motivating Taylor Polynomials 1We motivate the Taylor coefficient formula by showing how to recover a polynomial from its derivatives at a point. Lecture 35 Play Video Motivating Taylor Polynomials 2Calculus: 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 Play Video Application of Taylor Series: Re-centering PolynomialsWe 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 Play Video Approximating with Maclaurin PolynomialsWe 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 Play Video Approximating with Taylor PolynomialsWe 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 Play Video Fast Maclaurin Polynomial for Rational FunctionCalculus: 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 Play Video Taylor's Theorem for RemaindersCalculus: 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 Play Video 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 Play Video Power Series 1a - Interval and Radius of ConvergenceWe 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 Play Video Power Series 1b - Interval of Convergence Using Ratio TestFor 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 Play Video Example of Interval of Convergence Using Ratio TestCalculus: 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 Play Video Power Series 1c - Interval of Convergence Using Root TestWe 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 Play Video Power Series 1d - Finding the CenterWe 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 Play Video Power Series with SquaresWe 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 Play Video Derivative/Antiderivative of a Power Series 1a - BasicsWe 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 Play Video Derivative/Antiderivative of a Power Series 1b - Interval of ConvergenceWe 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 Play Video Derivative/Antiderivative of a Power Series 1c - More ExamplesWe 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 Play Video Increasing the Interval of ConvergenceWe 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 Play Video Constructing Power Series from Functions 1a - Geometric Power SeriesWe 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 Play Video Constructing Power Series from Functions 1b - More Geometric Power SeriesContinuing 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 Play Video Constructing Power Series from Functions 1c - Taylor CoefficientsCalculus: 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 Play Video The Taylor Series for f(x) = ln(x) at x = 1Calculus: 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 Play Video The Maclaurin Series for f(x) = 1/(1-x)^2Calculus: 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 Play Video The Maclaurin Series for f(x) = e^xCalculus: We derive the Maclaurin series for e^x and estimate e^{-1} by way of the error estimate for alternating series. Lecture 58 Play Video 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 Play Video The Maclaurin Series of f(x) = (1+x)^{1/2} 1aCalculus: 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 Play Video The Maclaurin Series for f(x) = (1+x)^{1/2} 1bCalculus: 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.