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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

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(%);

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