Home ## The Fundamental Theorem of CalculusThree Different ConceptsThe Fundamental Theorem of Calculus (Part 2) The Fundamental Theorem of Calculus (Part 1) More FTC 1 ## The Indefinite Integral and the Net ChangeIndefinite Integrals and Anti-derivativesA Table of Common Anti-derivatives The Net Change Theorem The NCT and Public Policy ## SubstitutionSubstitution for Indefinite IntegralsExamples to Try Revised Table of Integrals Substitution for Definite Integrals Examples ## Area Between CurvesComputation Using IntegrationTo Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary ## VolumesSlicing and Dicing SolidsSolids of Revolution 1: Disks Solids of Revolution 2: Washers More Practice ## Integration by PartsIntegration by PartsExamples Integration by Parts with a definite integral Going in Circles Tricks of the Trade ## Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric FunctionsProduct of Sines and Cosines (mixed even and odd powers or only odd powers) Product of Sines and Cosines (only even powers) Product of Secants and Tangents Other Cases ## Trig SubstitutionsHow Trig Substitution WorksSummary of trig substitution options Examples Completing the Square ## Partial FractionsIntroductionLinear Factors Irreducible Quadratic Factors Improper Rational Functions and Long Division Summary ## Strategies of IntegrationSubstitutionIntegration by Parts Trig Integrals Trig Substitutions Partial Fractions ## Improper IntegralsType 1 - Improper Integrals with Infinite Intervals of IntegrationType 2 - Improper Integrals with Discontinuous Integrands Comparison Tests for Convergence ## Differential EquationsIntroductionSeparable Equations Mixing and Dilution ## Models of GrowthExponential Growth and DecayLogistic Growth ## Infinite SequencesApproximate Versus Exact AnswersExamples of Infinite Sequences Limit Laws for Sequences Theorems for and Examples of Computing Limits of Sequences Monotonic Covergence ## Infinite SeriesIntroductionGeometric Series Limit Laws for Series Test for Divergence and Other Theorems Telescoping Sums and the FTC ## Integral TestRoad MapThe Integral Test Estimates of Value of the Series ## Comparison TestsThe Basic Comparison TestThe Limit Comparison Test ## Convergence of Series with Negative TermsIntroduction, Alternating Series,and the AS TestAbsolute Convergence Rearrangements ## The Ratio and Root TestsThe Ratio TestThe Root Test Examples ## Strategies for testing SeriesStrategy to Test Series and a Review of TestsExamples, Part 1 Examples, Part 2 ## Power SeriesRadius and Interval of ConvergenceFinding the Interval of Convergence Power Series Centered at $x=a$ ## Representing Functions as Power SeriesFunctions as Power SeriesDerivatives and Integrals of Power Series Applications and Examples ## Taylor and Maclaurin SeriesThe Formula for Taylor SeriesTaylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Useful Facts ## Applications of Taylor PolynomialsTaylor PolynomialsWhen Functions Are Equal to Their Taylor Series When a Function Does Not Equal Its Taylor Series Other Uses of Taylor Polynomials ## Partial DerivativesVisualizing Functions in 3 DimensionsDefinitions and Examples An Example from DNA Geometry of Partial Derivatives Higher Order Derivatives Differentials and Taylor Expansions ## Multiple IntegralsBackgroundWhat is a Double Integral? Volumes as Double Integrals ## Iterated Integrals over RectanglesHow To Compute Iterated IntegralsExamples of Iterated Integrals Cavalieri's Principle Fubini's Theorem Summary and an Important Example ## Double Integrals over General RegionsType I and Type II regionsExamples 1-4 Examples 5-7 Order of Integration |
## The Ratio Test
Let $\sum a_n$ be a series. The
Notice that the Ratio Test considers the ratio
of the absolute values of the terms. As you
might expect, the Ratio Test thus gives us information about whether
the series $\sum a_n$ converges absolutely.
Warning: There are
examples with $L=1$ that converge absolutely, examples that converge
conditionally, and examples that diverge. DO: Apply the Ratio Test to 1) the
absolutely convergent series $\sum\frac{1}{n^2}$, 2) the
conditionally convergent series $\sum-\frac{1}{n}$, and 3) the
divergent series $\sum\frac{1}{n}$.1) As $n\to\infty$, $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{(n+1)^2}}{\frac{1}{n^2}}=\frac{n^2}{(n+1)^2}\longrightarrow 1$. 2) As $n\to\infty$, $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$. 3) As $n\to\infty$, $\displaystyle\frac{\left|a_{n+1}\right|}{\left|a_n\right|}=\frac{\frac{1}{n+1}}{\frac{1}{n}}=\frac{n}{n+1}\longrightarrow 1$. We used other tests to determine the convergence/divergence of these series - the Ratio Test fails
to help us with these series.DO: The only way a series could
be conditionally convergent is if the Ratio Test fails for that
series. Why?## Review of simiplificationAs you work through this module, you must be able to work with ratios of factorials as well as ratio of powers. Recall that $n!=1\cdot 2\cdot 3\cdots(n-1)\cdot n$.DO:
Simplify $\frac{(n+1)!}{n!}$.$\frac{(n+1)!}{n!}=\frac{1\cdot 2\cdot 3\cdots(n-1)\cdot n\cdot (n+1)}{1\cdot 2\cdot 3\cdots(n-1)\cdot n}=n+1$ after all the cancellation. DO: Simplify
$\displaystyle\frac{\frac{50^{n+1}}{(n+1)!}}{\frac{50^n}{n!}}$$\displaystyle\frac{\frac{50^{n+1}}{(n+1)!}}{\frac{50^n}{n!}}=\frac{50^{n+1}}{(n+1)!}\cdot\frac{n!}{50^n}=\frac{50^{n+1}}{50^n}\cdot\frac{n!}{(n+1)!}=50\cdot\frac{1}{n+1}=\frac{50}{n+1}$ A couple of worked out examples of the Ratio Test are contained in the video, as well as the ideas of why the Ratio Test works. |