陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Another problem about power series」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured.

Problem 1 Let be a formal function of one variable . Suppose that is the formal function defined by

已知结果和反例

where we use to denote the -fold derivative of with respect to the variable .

To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily define

证明或构造的主线

To illustrate Problem 1 (i), let us compute up to third order in , using to denote any quantity involving four or more factors of and its derivatives, and similarly for other exponents than . Then we have

Thus Problem 1 (i) holds up to errors of at least. In principle one can continue verifying Problem 1 (i) to increasingly high order in , but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.

阅读时建议盯住的点

Problem 1 (i) can also be posed in formal power series: if

is a formal power series with no constant term with complex coefficients with , then one can verify that the series

值得单独记下的条目

• (i) Show that can be formally recovered from by the formula
• (ii) There is a remarkable further formal identity relating with that does not explicitly involve any infinite summation. What is this identity?

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源