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
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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:
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:
Maple gives this value for the integral:
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