Taylor Polynomial Error for x^(1/3) around x=8

When we consider the error in a Taylor polynomial (finite) approximation, there are two factors to consider:

the degree of the polynomial we will use (higher degree is more accurate, but takes longer to compute), and
the specific x value where want to approximate sin(x); closer to the x=a center is better, further out the approximation will get worse.

The bound on the Taylor Polynomial error can be computed in one of two ways:

If the series is alternating for the given x value, we can use the Alternating Series Error Bound of the first omitted term.
If the series is not alternating for the given x value, we have to use Taylor's inequality, which is a little more complicated.

For the power series for

, the series is alternating when x > 8, but not when x < 8. That means - for x>8, the first omitted term is large enough error to include the original function, but - for x<8, the first omitted term is not large enough error, and we need to use the larger Taylor Inequality error.

Taylor Polynomial Error for x^(1/3) around x=8

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