Jozef

### Questions (166)

 32 Prove: $\int_0^\infty \sin (x^2) \, dx$ converges. 30 Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$ 27 Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$ 19 Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$ 18 Does $\int_{1}^{\infty}\sin(x\log x)dx$ converge?

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 3 Reduction from Hamiltonian cycle to Hamiltonian path 1 $f,g: \mathbb{R} \to \mathbb{C}, 2\pi$ periodic. Prove: if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$ 0 $A,B,C \in M_{n} (\mathbb C)$ and $g(X)\in \mathbb C[x]$ such that $AC=CB$- prove that $A^jC=CB^j$ and $g(A)C=Cg(B)$

### Tags (57)

 3 computer-science × 16 0 ordinary-differential-equations × 14 3 graph-theory × 6 0 integration × 13 1 calculus × 81 0 real-analysis × 11 1 fourier-analysis × 7 0 linear-algebra × 11 0 elementary-number-theory × 24 0 convergence × 8