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 $x <br /> eq 0$
  • $f(0) = 0$
• 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:
  • $R<em>n(x) = f(x) - T</em>n(x)$ (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.