Taylor Series Expansion Example Report

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    : Taylor Series Expansion
    Notes: Tap here first...

    Series expansion is a common mathematical tool for approximating functions that cannot be directly
    calculated.  This example will show how Math Minion's matrix object can be used to easily implement
    a series expansion, specifically a Taylor series to calculate e^x and sin(x).

    See the notes for each of the objects.
    Expression: Root.x
    Notes: Just an arbitrary value for the functions to work on.
    Formula: 2
    Unit:  Fraction
        1        2.00000

    Matrix: Root.taylorE
    Notes: An exponential function e^x can be represented by the Taylor series:

    e^x = sum from n=0 to infinity of:

            x^n / n!

    The terms of this summation, except for n=0, are represented in this matrix as a column calculated
    by the formula

    x^{row}/{factorial {row}}

    which is assigned to the column header cell of column 1.

    The function {row} will return the number of each row and thus represents n starting from 1 rather
    than 0.  This means we still have to add the n=0 term, which is just 1, when we sum the terms.

    The number of terms is set by setting the number or rows in the matrix.
    Input Sources: x
    Cell Formulas:
    0_0: x^{row}/{factorial {row}}
    Unit: Fraction
        1        2.00000
        2        2.00000
        3        1.33333
        4        0.66667
        5        0.26667
        6        0.08889
        7        0.02540
        8   6.349206e-03
        9   1.410935e-03
    10   2.821869e-04

    Expression: Root.sumE
    Notes: Using the sum function with TaylorE as the argument will sum all the terms of the expansion
    except for the n=0 term.

    The value of this term is just 1, and it is added to the function for a formula:

    1+{sum taylorE}

    Input Sources: taylorE
    Formula: 1+{sum taylorE}
    Unit:  Fraction
        1        7.38899

    Expression: Root.diffE
    Notes: Math Minion has an e to the x function, so we can easily check the accuracy of our
    approximation with the formula:

    {exp x} - sumE

    Try altering the number of terms in the expansion to see how that affects the accuracy.  This is
    done by just changing the number of rows in the TaylorE matrix object.
    Input Sources: sumE x
    Formula: {exp x} - sumE
    Unit:  Fraction
        1   6.138994e-05

    Matrix: Root.rowE
    Notes: In order to plot the convergence as the number of terms increase, the formula for the column
    in this matrix:

    1+{sum taylorE[1:{row}]}

    Is similar to the one for sumE, but it uses the range operator : and the index operators [] to sum
    the first n rows of taylorE.

    Note that the number of rows for this matrix is set to the formula:

    {nrows taylorE}

    so that it is always the same size as taylorE.

    Input Sources: taylorE
    Cell Formulas:
    0_0: 0
    0_1: 1+{sum taylorE[1:{row}]}
    Unit: Fraction
        1        3.00000
        2        5.00000
        3        6.33333
        4        7.00000
        5        7.26667
        6        7.35556
        7        7.38095
        8        7.38730
        9        7.38871
    10        7.38899

    Graph/Table: Root.Plot
    Notes: This plots the sum to n of the taylorE terms, as calculated by rowE, versus n as calculated
    by the formula:

    1:{nrows taylorE}

    Note that term 0, which would be 1, is not plotted.

    Input Sources: rowE taylorE
    1:{nrows taylo           rowE
        Fraction       Fraction
        1.00000        3.00000
        2.00000        5.00000
        3.00000        6.33333
        4.00000        7.00000
        5.00000        7.26667
        6.00000        7.35556
        7.00000        7.38095
        8.00000        7.38730
        9.00000        7.38871
        10.00000        7.38899

    Matrix: Root.taylorSine
    Notes: This follows the same procedure as taylorE, but uses the formula:

    x^(2*{row}+1) * -1^{row}/{factorial 2*{row}+1}

    to calculate the Taylor expansion terms for  sin(x), which are:

    ((-1)^n) *  x^(2n+1)
    Input Sources: x
    Cell Formulas:
    0_0: x^(2*{row}+1) * -1^{row}/{factorial 2*{row}+1}
    Unit: Fraction
        1       -1.33333
        2        0.26667
        3       -0.02540
        4   1.410935e-03
        5  -5.130672e-05

    Expression: Root.sumSine
    Notes: In this case the n=0 term is just x, which is added to the sum of their other terms
    calculated by taylorSine.
    Input Sources: taylorSine x
    Formula: x+{sum taylorSine}
    Unit:  Fraction
        1        0.90930

    Expression: Root.diffSine
    Notes: Calculates the difference between the approximation and the built in sine function.
    Input Sources: sumSine x
    Formula: {sin x}-sumSine
    Unit:  Fraction
        1   1.290863e-06