# Random stuff 2026 week 1
Does this series converge or diverge, $\sum {2^{n^2}} / {n^{2^n}}$ ?
Cute little problem that can be done resolved with ratio test.
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Every rational number $q$ is a sum of three cubes of rationals.
This is not immediate to me how. But I will think about it later.
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A cute result, if $A,B$ are $2 \times 2$ matrices, then their commutator $C=[A.B]=AB-BA$ is such that $C^{2}$ is a diagonal matrix.
A hint: Think about the trace of $C$, and then apply Cayley-Hamilton to $C$.
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Given $a,b,c,d,e$ positive integers such that $a+b+c+d+e = abcde$, what is the maximum of $\max(a,b,c,d,e)$? Also, find all solutions to such $a,b,c,d,e$.
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A mother tells you that she has two children, and that one of them is a boy, what is the probability that the other child is also a boy?
Another mother tells you that she has two children, and one of them is a boy that is born on a Tuesday, what is the probability that the other child is also a boy?
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Optimal stopping time.
Say one tosses a fair coin, what is the expected number of tosses to see a particular sequence, say $HTTH$?
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Let $(a_{n})$ be a real sequence such that $a_{n}-a_{n+2} \to 0$. Show $\frac{a_{n}-a_{n+1}}{n}\to 0$.
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Multiplicative derivative
Define $f^{g}(x) = \lim_{h\to 0} (\frac{f(x+ h)}{f(x)})^{(1/h)}$, the geometric derivative of $f$ at $x$. What can we say about this operation?