12. Analysis and Approximation

12.1

Convergence of infinite series. Test for convergence: ratio test; limit comparison test; integral test.

See Monotonic Sequences, from Analysis WebNotes. The entry on monotonic sequences includes an interactive JAVA applet.

There is a short definition with an example at Definitions : Monotonic

Monotonicity is defined at Sequences, part of Interactive Real Analysis. Scroll down to the entry, which is followed by diagrams of monotonic, bounded sequences (below).

See Sequences, from Interactive Real Analysis, for an introduction to sequences and their convergence.

See the Convergence of Sequences entry from Analysis WebNotes. There is also an animated demo of convergence.

For a general discussion, see Sequences and Series, from the Oregon State University Math Department's Web Study Guide.

List of Series Tests, from the OSU Math Department's Web Study Guide, includes the divergence, integral, comparison, limit comparison, alternating series, absolute convergence, ratio and root tests for convergence.

For the root and ratio tests, see Tests for Convergence, from Analysis WebNotes.

The lecture notes on Taylor's Theorem, from MATA24S/25F/26Y, the home page of the first year calculus course at the University of Toronto, Scarborough College, include material on the ratio, root and integral tests, as well as some discussion of alternating series. This .pdf file takes several minutes to download. A free reader is available from Adobe Acrobat.

A monotonic, increasing sequence, bounded above. The diagram is taken from Sequences, part of the Interactive Real Analysis site from Seton Hall University. Used with permission.

12.2

Alternating series. Conditional convergence.

12.3

Power series: Radius of convergence. Determination of the radius of convergence by the ratio test.

12.4

Rolle's theorem; the mean value theorem. Applications of these theorems.

For an introduction th the mean value theorem, see The Mean Value Theorem, from CalculusQuest.

The mean value theorem is discussed at Differentiable Functions, part of the on-line interactive textbook Interactive Real Analysis from Seton Hall University.

Also see Theorem 6.15, from Analysis WebNotes. A proof is provided.

MATA24S/25F/26Y has extensive lecture notes on The Mean Value Theorem for Rational Functions and Rolle's Theorem, The Mean Value Theorem, and L'Hôpital's Rule, available as .pdf files. The files may take several minutes to download. A free reader for the files may be downloaded from Adobe Acrobat.

12.5

Use of Taylor series expansions, including the error term. Maclaurin series as a special case. Taylor polynomials. Taylor series by multiplication.

For introductory material on the Taylor series, see Taylor Series - Why We Want Them and How We Find Them, from James Sellers of the Undergraduate Computational Engineering and Sciences (UCES) Project.

Sequences and Series, from the Oregon State University Math Department's Web Study Guide, contains a good deal of information on Taylor series. More specifically, see Taylor Series, Error Bounds using Taylor Polynomials, and List of Common Maclaurin Series.

Calculus : An Overview discusses Taylor and Maclaurin Series as well as other topics from analysis and calculus.

Also see Taylor Series from the S.O.S. MATHematics site.

For a more advanced treatment, see Taylor Series, from Analysis WebNotes.

MATA24S/25F/26Y has sixteen pages of lecture notes on Taylor Approximations, and a further twenty-six pages of notes on Taylor's Theorem. The notes on Taylor approximations contain a discussion of the remainder in Taylor's formula. Both files take several minutes to download. A free reader for the files may be downloaded from Adobe Acrobat.

  
The convergence of the Taylor series for cos(x), centered on p/2. As the degree of the polynomial increases, its graph more closely resembles that of the cosine function.
Graphed using Mathematica.

12.6

Numerical integration. Derivation and application of the trapezium rule and Simpson's rule. The forms of the error terms; their use.

Introduction to Numerical Integration, from Chrisopher Johnson of the University of Utah's Hamlet Project, provides a thorough introduction to the subject.

At the Numerical Integration Tutorial, created by Joseph Zachary, also from the University of Utah, a JAVA applet makes it possible to display any one of a number of functions, and to calculate the area between the function, two moveable white lines, and the x-axis. Various numerical methods of calculation can be used, including the trapezeum method. The tutorial is a resource for the book Introduction to Scientific Programming. Recommended.

Visual Calculus provides illustrations of both the trapezium rule, at Trapezoidal Rule - 1,
and Simpson's Rule, at Simpson's Rule - 1.

Also see The Lower Riemann Sums demo (below), from John Orr. Recommended.

At the The Lower Riemann Sums demo, clicking on the x-axis causes green rectangles to be drawn. As the rectangles increase, the green area approaches the true area under the curve. At another demo, it is also possible to determine The Upper Riemann Sums. Used with permission.

12.7

The solution of non-linear equations by iterative methods, including the Newton-Raphson method; graphical interpretations. Fixed-point iteration; conditions for convergence. The concept of order of convergence (without proof).

At One-Dimensional Interation : The Quadratic Map, a JAVA applet is used to demonstrate iteration. When a point in the graph window is clicked, the iterates of that x value are shown. From The Geometry Center of the University of Minnesota.

For material related to iteration and its representation using Mathematica, Maple, and the TI-92 calculator, see Cobweb Diagrams, from The Connected Curriculum Project.

At Interactive Real Analyis there is a Root Finder (a JAVA applet) that finds zeros using either Newton's Method or the Bisection Method. Click Java Tools

MATA24S/25F/26Y, the home page of the first year calculus course at the University of Toronto, Scarborough College, has eight pages of lecture notes on Newton's Method.

Newton's Method : An Introduction has both an explanation and a dynamic demonstration of the process. See Newton's Method in Action -- An Animation. The site is part of The Connected Curriculum Project.

Using the TI-83 graphing calculator, iterations, including those based on the
Newton-Raphson method, can be both carried out and graphed, and their tables generated.

The positive root of x2 + 3x - 1 = 0 is approximated using the rearrangement x = (1 - x2)/3. The diagram shows the third iteration.

The iterations are summarized in a table. At the sixteenth iteration, the approximation is 0.3027756377, accurate to 10 decimal places.