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    Powere Series Expansion and its Utilization in Integration

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    Problem:
    Given the power series for the following function (1+x)^k

    (a) Write the power series for (1+x)^(1/3)
    (b) Use the power series from part (a) to find the power series for x^3
    (c) Using this series approximate the following integral (1+x^3) ^(1/3) using the first three terms

    © BrainMass Inc. brainmass.com December 24, 2021, 5:05 pm ad1c9bdddf
    https://brainmass.com/math/integrals/power-series-expansion-utilization-integration-26966

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    Note that

    .

    It is the same as the summation notation sigma, only instead of summing, we multiply the terms.

    For k=1/3 we get:

    Or in the compact form:

    If we use the change of variables:

    We see that:

    And we know that:

    It is easy to see that beyond the y3 term, all the terms will be zero, since one of the factors in the numerator will always be zero, hence:

    Substituting back y=x-1 we obtain:

    And indeed:

    To evaluate we will use again a substitution:

    And the integral becomes:

    Using the first three terms (see part a) we obtain:

    Substituting back we get:

    Thus:

    Maple gives this value for the integral:

    > Int((1+x^3)^(1/3),x=0..0.5);

    > evalf(%);

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:05 pm ad1c9bdddf>
    https://brainmass.com/math/integrals/power-series-expansion-utilization-integration-26966

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