Pure Mathematics F4: Power series

18 flashcards · Mathematics · by lakesare

1. [Taylor Polynomial] Visually imagine the Taylor Polynomial formula Tn(x).

   n - degree of the Taylor Polynomialf - the function we are approximatinga - at what point are we approximating this function

2. [Taylor Polynomial] How did we determine the formula for the Taylor Polynomial? n - degree of the Taylor Polynomialf - the function we are approximatinga - at what point are we approximating this function

   The coefficients in the definition of Tn have been chosen so that each derivative of f at a is equal to derivative of Tn at a.Thus the functions f and Tn have the same value at a and have equal derivatives at a for all orders up to and including n.

3. [Taylor's Theorem] Visually imagine Taylor's Theorem.

   f(x) - function we're approximatingTn(x) - Taylor PolynomialRn(x) - Remainder term/Error term

4. [Taylor Polynomial, Taylor's Theorem] Taylor Polynomial VS Taylor's Theorem?

   Taylor Polynomial Taylor's Theorem

5. [Taylor Polynomial] How do these rules lead to the definition of the Taylor polynomial?

   Tn(n)(a) kills terms to the left Tn(n)(a) kills terms to the right

6. Name 3 theorems in terms of generalization: Rolle's Theorem _________________ _________________

   Rolle's Theorem Mean Value Theorem Taylor's Theorem

7. [Taylor's Theorem] Visually imagine Taylor's Theorem for n=0.

   Thus Taylor's Theorem can be considered as a generalisation of the Mean Value Theorem.

8. [Taylor's Theorem] How would we show how precise: the approximation of f(x) at x by the Taylor polynomial?

   Find |Rn(x)| Find Remainder term/Error termUse the knowledge that c ∈ [a, x] interval I

9. [Taylor Polynomial vs Taylor Series] What's Taylor Polynomial VS Taylor Series?

   Taylor Polynomial - Tn(x) Taylor Series - limn → ∞ Taylor Polynomial

10. [Taylor Polynomial] Name 2 ways to use Taylor Polynomials.

    approximation of a function f(x) rewriting of a function f(x) In this module, sin x and cos x are defined in terms of a right-angled triangle. This theorem shows that sin x and cos x can be represented by power series for all x ∈ R, and some texts use these series to define sin x and cos x in a way that does not depend on geometric ideas.Also, some texts define ex using this power series.

11. [Power Series] Visually imagine Power Series.

    Notice that in the definition of a power series, we think of a as a constant and x as a variable.All Taylor series are examples of power series.

12. [Power Series] Visually imagine the centre of the Power Series.

13. [Taylor Series] When are the Taylor Series said to be valid?

    When Taylor Series exactly = f(x).

14. [Power Series] Visually imagine the Radius of Convergence Theorem.

    Σ converges: one point (a) everywhere interval [a - R, a + R] Theorem F66 makes no assertion about the behaviour of the power series at the endpoints of the interval (a − R, a + R); in fact, a power series may converge at both endpoints, neither endpoint or exactly one endpoint.

15. [Power Series] Radius of convergence VS interval of convergence?

    Radius of convergence - R Interval of convergence - ([a - R, a + R])

16. [Power Series] What test do we usually use to find the interval of convergence for Power Series?

    Ratio Test

17. Binomial Theorem VS General Binomial Theorem?

    Binomial Theorem - (a + b)natural or (1 + x)natural General Binomial Theorem - (1 + x)real Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers.

18. [Taylor Series] Name 3 ways to find Taylor series.

    Taylor's Theorem Combination Rules Differentiation and Integration Rules