Taylor Series and Remainder Estimates

Difference Between Taylor Function and Function Accuracy

• Understanding the Taylor polynomial accuracy is vital in numerical approximation.
• The Taylor remainder estimate indicates the bound for this error, helping to ensure closeness between the Taylor polynomial and the true function value.
• Examples: Taylor polynomials can approximate functions such as e^x, ext{sine}(x), and ext{ln}(1+x) with known accuracy depending on the number of terms used.

Pathological Example: Function f(x) Definition

• Define the function:
  • f(x) = e^{-1/x^2} if x
    eq 0
  • f(0) = 0
• The Maclaurin series for this function is only equal to the function at the point x=0, illustrating that a function's Taylor series may converge only at a single point despite derivatives being nonzero.
  • All derivatives at x=0 are zero, leading to the conclusion that the Maclaurin series produces a flat line (zero series) at zero.
  • Importance: This example emphasizes the necessity for a remainder estimate when applying Taylor series, as it shows that convergence does not imply equality with the original function.

Taylor Remainder Estimate

• Remainder defined as:
  • Rn(x) = f(x) - Tn(x) (where T_n(x) is the nth degree Taylor polynomial)
• To confirm equivalence of the Taylor series and the function, the limit of the remainder must go to zero across the interval of convergence.
• If the remainder is bounded and approaches zero, then the Taylor series converges to the function:
  • |R_n(x)| ext{ is bounded by } \frac{m}{(n+1)!} |x-a|^{n+1}, where m relates to the (n+1)th derivative on the interval.

Visualizing the Remainder and Finding Bounds

• Establish intervals centered at a point a and consider the distance d in both directions.
• Finding an upper bound on the absolute value of the (n+1)th derivative will determine the remainder's behavior as n approaches infinity.

Example: Proving Equality for e^x

• To show that e^x equals its Maclaurin series, identify a specific value of x.
• Use Taylor's inequality to show:
  • Define d such that |x| ext{ is within } d.
  • With e^{(x)} ext{ being bounded by } e^d, apply Taylor's inequality to conclude that the remainder R_n(x) approaches zero as n increases.

Example: Proving for ext{sine}(x)

• f(x) = ext{sine}(x) has its derivatives bounded by 1, which allows setting m=1 in Taylor's inequality.
• Again, establish that the remainder R_n(x) also approaches zero, confirming the series represents the function for all x:
  • |R_n(x)| ext{ convergek to } \frac{|x|^{n+1}}{(n+1)!}.

Calculation the Taylor Series Centered at rac{
ext{pi}}{3}

• Follow similar procedures to compute the derivatives of ext{sine}(x); evaluate derivatives at rac{
  ext{pi}}{3}.
• Create the Taylor series based on computed values, noting a repeating pattern every four derivatives.
• Confirm equivalence by showing that R_n(x) goes to zero for all x considering bounded derivatives.

Upcoming Questions on Taylor Approximations

• Future lessons will include assessing Taylor polynomials for specific functions and intervals, looking particularly at accuracy, error bounds, and utilizing these estimations to derive accurate evaluations at particular angles (e.g., 12 degrees in sine approximation).
• Expect to tackle exercises related to function approximations and how to determine the adequacy of Taylor series for specific applications.