Calculus Volume 2 · 免费预览
幂级数的性质
在区间内部，幂级数可以逐项求导和逐项积分，收敛半径保持不变，从而由已有级数生成新的级数，并以级数方法求解微分方程。
1. Term-by-term differentiation
Term-by-term differentiation
2. Term-by-term integration
Term-by-term integration
3. Algebra of series
Algebra of series
4. Series solutions of differential equations
Series solutions of differential equations
5. Differentiation effect on radius
Term-by-term differentiation of a power series does what to its radius of convergence?
6. Geometric derivative
Differentiating the geometric series sum of x^n equals 1/(1 - x) gives which sum?
7. Logarithm series
Integrating the geometric series and substituting -x gives ln(1 + x) equals which series?
8. Leibniz series value
The arctangent series at x = 1 gives Leibniz's formula for pi. What is the value of the series 1 - 1/3 + 1/5 - ...?
9. Arctangent series origin
The arctangent series sum of (-1)^n x^(2n+1)/(2n+1) is obtained by integrating which function's series?
10. Integrated coefficient
Integrating a power series term c_n x^n gives c_n/(n+1) x^(n+1). As a single number, what divides the coefficient? Answer with a number.
11. Differentiated exponent
Differentiating c_n x^n gives n c_n x^(n-1). As a number, what is the new coefficient's factor? Answer with a word like n.
12. Product operation name
Multiplying two power series term by convolved term is called the Cauchy what? Answer with one word.
13. Cauchy product purpose
The Cauchy product of two power series is used to do what?
14. Airy equation form
Airy's equation y double prime - x y = 0 is solved by power series with a coefficient recurrence. As a single letter, what is the unknown function? Answer with one lowercase letter.