Calculus Volume 2 · 免费预览
泰勒级数与麦克劳林级数
函数在某点处的泰勒级数把各阶导数组装成系数，即 c_n 等于第 n 阶导数除以 n 的阶乘；多项式逼近被精化到无穷阶，并以余项的极限来验证展开式的有效性。
1. The coefficient formula
The coefficient formula
2. The remainder theorem
The remainder theorem
3. Binomial series and applications
Binomial series and applications
4. Calculating with series
Calculating with series
5. Coefficient formula
The coefficient c_n of the Taylor series of f at a equals which expression?
6. Maclaurin definition
A Taylor series centered at a = 0 is called what?
7. Remainder limit
The Taylor series of f equals f(x) exactly when which condition holds?
8. Binomial exponent range
The binomial series for (1 + x)^k converges for |x| less than 1 for non-integer k, and at x = 1 converges when which condition holds?
9. Cosine series
The Maclaurin series of cos x is sum of (-1)^n x^(2n)/(2n)!. Which feature of the cosine's derivatives produces the even powers only?
10. Exponential Maclaurin constant
Every derivative of e^x at 0 equals 1, so the Maclaurin coefficient c_n equals 1 over what function of n? Answer like n factorial.
11. Sine Maclaurin first term
The Maclaurin series of sin x begins with the term x. As a power, what is its exponent? Answer with a number.
12. Remainder factorial
The Lagrange remainder divides by (N + 1) factorial. As a number, how many more derivative orders than N does it involve? Answer with a number.
13. Remainder bound for e^x
For the exponential's Maclaurin series, the remainder is at most e^|x| times |x|^(N+1) over (N + 1) factorial. Why does this tend to zero?
14. Binomial coefficient recursion
The generalized binomial coefficient binom(k, n) satisfies binom(k, n+1) equals binom(k, n) times (k - n) over (n + 1). As a number, what is the denominator? Answer with a number.