Jozef

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162 Questions

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21	Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
12	Does $\int_{1}^{\infty}\sin(x\log x)dx $ converge?
12	Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.
11	Why does $\int_{0}^{\infty}\frac{dx}{1+(x \sin x)^2}$ diverge?
10	Prove: $\int_0^1 \frac{\ln x }{x-1}=\sum_1^\infty \frac{1}{n^2}$

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1,975 Reputation

+5	Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
+45	Prove by Hilbert deduction: $\vdash _{HFOL} \forall x (\neg(A \to \bar{B}))\to \neg(\forall xA \to \neg(\forall xB))$
+5	Prove: $f(x)$ continuous function and $\int_a^\infty \|f(x)\|\;dx < \infty$ so $\lim_{ x \to \infty } f(x)=0$
+5	$f(x)$ monotonic integrable function and $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$, Prove: $\lim_{x\to \infty}f(x)=a$

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3 Answers

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1	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 $g(A)C=Cg(B)$

42 Tags

1 calculus × 83	0 differential-equations × 14
1 computer-science × 16	0 linear-algebra × 11
1 fourier-analysis × 7	0 computability × 8
1 graph-theory × 6	0 real-analysis × 8
0 elementary-number-theory × 25	0 integral × 8

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$Mathematics Stack Exchange$	Mathematics	1,975 rep	225
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	Stack Overflow	144 rep	4
	Theoretical Computer Science	143 rep	4
	Electrical Engineering	107 rep	4

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