Approximation of functions:

• Taylor Series
• Fourier Series
• Convergence of Fourier Series
• Bernstein polynomials and polynomial approximation

The following Maple worksheets can be downloaded.

They are all compatible with Classic Worksheet Maple 10.

Taylor polynomials and Taylor series - taylor.mws

• Some polynomial approximations for exp(x) when x is close to zero
• General Taylor polynomial approximations for exp(x) when x is close to zero
• Linear approximations
• Quadratic approximations
• General Taylor polynomials and Taylor series

Maclaurin series and convergence of power series - maclrn.mws

• Power series
• Radius and interval of convergence of power series
• Convergence of the Maclaurin series for sin(x)
• Some standard Maclaurin series and their intervals of convergence
• Why does a power series have a radius of convergence?

Experimental determination of the radius of converge - radconv.mws

• Experimental determination of the radius of convergence of a Maclaurin series
• Examples

Using Taylor series to approximate integrals - taylrint.mws

• Introductory example
• More examples 

Using Taylor series to calculate pi  - calcpi.mws

• Newton's approximation for pi
• Using the Maclaurin series for arctan(x) to calculate pi
• 19 th century computation of pi by Johann Dase

Manipulation of of power series - diffpow.mws

• Introductory examples of adding power series
• Formal addition of power series
• Multiplying a power series by a polynomial
• Differentiating power series
• Integrating power series

The even and odd parts of a function - evenodd.mws

• Even and odd functions
• The even and odd parts of a function

Hyperbolic functions - hyperbolic.mws

• The hyperbolic sine and cosine functions
• Two points on the graph of cosh and sinh

Multiplication of power series - multpow.mws

• From multiplying polynomials to multiplying power series
• Examples of multiplying power series

Bernoulli numbers and the Maclaurin series for x/(exp(x)-1) - bernum.mws

• An introduction to Bernoulli numbers via the sums of powers of positive integers
• The Bernoulli numbers via the series for x/(exp(x)-1)
• A summation formula involving the Bernoulli numbers
• Calculation of the Bernoulli numbers
• Procedures for computing the Bernoulli numbers
• Ada Lovelace and the first computer program

Euler numbers and the Maclaurin series for sech(x) - eulnum.mws

• The Euler numbers
• A summation formula involving the Euler numbers
• Calculation of the Euler numbers
• Procedures for computing the Euler numbers

Radius of convergence of the series for x/(exp(x)-1) and sech(x) - maclrn2.mws

• The Maclaurin series for x/(exp(x)-1)
• The Maclaurin series for sech(x)

Some Maclaurin series involving Bernoulli and Euler numbers - maclrn3.mws

• The Maclaurin series for coth(x)-1/x
• The Maclaurin series for tanh(x)
• The Maclaurin series for tan(x)
• The Maclaurin series for sec(x)

More Maclaurin series involving Bernoulli numbers - maclrn4.mws

• The Maclaurin series for x/sinh(x)
• The Maclaurin series for x/sin(x)
• The Maclaurin series for ln(sinh(x)/x)
• The Maclaurin series for ln(sin(x)/x)

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