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### Questions (42)

 16 How to analyze $\sup_{x>0}|e^xf(x)| < \infty$ and $\sup_{n\in\mathbb{N}} |f^{(n)}(0)|< \infty$? 15 Can we divide $\mathbb{R}^2$ into two connected parts such that each part is not simply-connected? 8 Evaluating $\int_0^\infty \left( \frac{x}{e^x-e^{-x}}-\frac{1}{2} \right) \frac{dx}{x^2}$ 8 Evaluating $\int_0^{\pi/2} \log \left| \sin^2 x - a \right|$ where $a\in [0,1]$. 8 Proving $\int_0^\infty \log\left (1-2\frac{\cos 2\theta}{x^2}+\frac{1}{x^4} \right)dx =2\pi \sin \theta$

### Reputation (920)

 +10 From $\|g\|_2 =1$ to $\|g\|_\infty^2 \ge \dim V$ on a subspace of $C[0,1]$ +35 From $\|g\|_2 =1$ to $\|g\|_\infty^2 \ge \dim V$ on a subspace of $C[0,1]$ +5 Evaluating $\int_1^\infty \sin \frac{1}{x^2} dx$ +5 How to analyze $\sup_{x>0}|e^xf(x)| < \infty$ and $\sup_{n\in\mathbb{N}} |f^{(n)}(0)|< \infty$?

 2 Closed form representation of alternating series 1 From $\|g\|_2 =1$ to $\|g\|_\infty^2 \ge \dim V$ on a subspace of $C[0,1]$ 1 Find a set on which all the $f_i$ have the same integral. 1 How to calculate $\lim_{n \to \infty} \sum_{k=1}^{n}\frac{1}{\sqrt {n^2+n-k^2}}$? 1 How to analyze $f(f(x))=-x^3+\sin(x^2+\ln(1+\left|x\right|))$?

### Tags (43)

 3 real-analysis × 32 0 definite-integrals × 5 2 sequences-and-series × 8 0 number-theory × 4 2 limits × 6 0 matrices × 4 1 calculus × 11 0 complex-analysis × 4 0 linear-algebra × 6 0 trigonometry × 4

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