18 Why does this pattern in powers happen? 12 What is the period of Langton's ant on a torus? 12 Does the infinite nested logarithm $\ln(2\ln(3\ln(4\ln(5\ln(6...)))))$ converge? 8 Show that $\mathbb{R}^{n}$ ($n>2$) minus a countable number of lines is still path-connected 7 Arrange numbers 1-12 around a circle so that any three consecutive numbers have a sum $\leq 20$?

### Reputation (6,716)

 +20 Give a combinatorial proof for $n^2 = n(n-1)+n$ +25 Characteristic polynomial of an $n \times n$ matrix with only one column and one row non zero. +35 Find all natural n such that gcd(n, 1260)=70 and the number of positive divisors of n is 30 +10 Find all possible values of $f(2018)$

### Questions (26)

 10 Set of elements of degree $2^n$ over a base field is itself a field 9 Closed form for $\sum_{k=0}^\infty \frac{x^k}{(k+1)^4 k!}$ 8 Does $\lim_{x \to \infty} f(x) = 0$ imply $\int_\delta^\infty \frac{f(x)}{x}\, dx$ converges? 7 Fibration over contractible space is homotopic to a fiber 6 Basis for $\mathbb{R}^\mathbb{N}$ implies axiom of choice?

### Tags (204)

 61 calculus × 39 34 sequences-and-series × 15 45 combinatorics × 15 32 convergence-divergence × 7 39 elementary-number-theory × 15 26 functions × 8 38 real-analysis × 25 25 algebra-precalculus × 12 38 number-theory × 13 23 integration × 12

### Bookmarks (24)

 48 'flimsy' spaces: removing any $n$ points results in disconnectedness 43 What is $\tau(A_n)$? 38 What is the period of Langton's ant on a torus? 36 New bound for Am-Gm of 2 variables 31 Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+}$ such that $\frac{a^2+b}{a+b^2}=n.$