Calculus Volume 2 · 免费预览
泰勒级数的实际运算
对泰勒级数作乘法、除法和复合，可以构造新的展开式、给出带误差保证的近似，并用于求取极限，这些正是级数实际应用的操作工具箱。
1. Composition and substitution
Composition and substitution
2. Multiplication and division
Multiplication and division
3. Approximations with error certification
Approximations with error certification
4. Limits by series
Limits by series
5. Substitution for e^(-x^2)
Which operation produces the series for e^(-x^2) from the exponential's Maclaurin series?
6. Tangent series method
The series for tan x is obtained by which manipulation of the sine and cosine series?
7. Series limit method
To evaluate lim x to 0 of (e^x - 1 - x)/x^2 by series, what does one read from the exponential's expansion?
8. Machin formula purpose
Machin's formula pi/4 = 4 arctan(1/5) - arctan(1/239) was historically used for what?
9. Reciprocal series condition
A reciprocal series 1/f(x) expands in powers of x at the origin when which condition holds?
10. Cauchy product coefficient
The Cauchy product of sum a_n x^n and sum b_n x^n has x^n coefficient sum over k of a_k b_(n-k). As a single letter, what is the summation index? Answer with a lowercase letter.
11. Substituted radius
Substituting 2x into a series of radius R gives the new radius R/2. For R = 2, what is the new radius? Answer with a number.
12. Division requirement
A reciprocal series 1/f expands at 0 when f(0) is nonzero. As a single word, what condition? Answer in lowercase.
13. Composing series difficulty
Composing two series, expanding f(g(x)) as a single power series, is best described as what?
14. Error function seed
The error function's series comes from integrating e^(-x^2), whose expansion is the exponential series with -x^2 substituted. As a two-word name, what function results? Answer like answer key.