§8.6 Quantifying the accuracy of approximations

Introduction

Alternating series of real numbers

Error Approximations for Taylor Polynomials

Summary

Determine how well the $100^{\text{th}}$ partial sum $S_{100}$ of

approximates the value of the converging alternating series.

Determine the maximum error possible when using the degree $10$ Taylor polynomial centered at $a = 0$ for $\sin(x)$ to approximate the value of $\sin(2)$.

For a convergent alternating series with decreasing terms, the error after summing $n$ terms is bounded by

Using $\sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$, the estimate $\sin(1) \approx 1-\frac{1}{3!}+\frac{1}{5!}-\frac{1}{7!}$ is guaranteed accurate to within

If $|f^{(n+1)}(x)| \le M$ on the interval between $a$ and $x$, the Lagrange error bound states $|f(x)-T_n(x)| \le$

Using $\cos(x^2) = 1-\frac{x^4}{2!}+\frac{x^8}{4!}-\cdots$, the series estimate of $\int_0^1\cos(x^2)\,dx$ through the $\frac{1}{13\cdot6!}$ term is $\frac{25399}{28080}$. Report this value, rounded to four decimals.

Compute the alternating estimate $1-\frac{1}{3!}+\frac{1}{5!}-\frac{1}{7!}$ for $\sin(1)$, rounded to four decimals.

At the endpoint $x = 1$, the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$ converges by the Alternating Series Theorem, and its sum is exactly $\ln(2)$.