Find the first four nonzero terms of the Taylor series for *f*(*x*) = *x*^{1/3} centered at *x* = 1.

Answer

Let's hope that none of the terms are zero, and find the degree 3 Taylor polynomial centered at 1 (since the degree *n* polynomial has *n* + 1 terms). We find derivatives, evaluate them at 1, and plug the resulting values into the magic formula. Since 1 raised to any power is 1, these aren't quite as messy as they appear at first. As with previous examples, we leave terms in expanded form to make the patterns easier to see.

Putting this into the formula, we get

Since none of the terms are zero, we've successfully found the first four nonzero terms. A graph confirms that our answer makes sense: