Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
Example 3.3.3
For with in Taylor's Formula, obtain the general form of .
Solution
Define the function
Type Context Panel: Assign Function
Obtain for
Notation for kth derivative evaluated at : Context Panel: Sequence≻ Choose to
Infer
The cyclic nature of the values of the derivatives at suggest only odd-powers of have nonzero coefficients, and these coefficients are either divided by the factorial of an odd integer. These observations suggest
Corroborate with a specific Taylor polynomial
Type
Context Panel: Series≻Series≻
Verify
Context Panel: Series≻Formal Power Series See Figure 3.3.3(a) for details.
A Formal Power Series is an infinite sum, the truncation of which is then just a polynomial.
The general term in the formal power series is the general term in , the Taylor polynomial of degree .
Figure 3.3.3(a) Formal Power Series dialog
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