Septimiu Cristian

### Questions (38)

 6 Show that ${\frac{x_1^{n}+x_2^{n}+…+x_n^{n}}{x_1x_2…x_n} + \frac{\sqrt[n]{x_1x_2…x_n}}{x_1+x_2+…+x_n}}\ge{n+\frac{1}{n}}$ 5 Having the sequence $(a_{n})_{n\geq1}$, $a_{n}=\int_{0}^{1} x^{n}(1-x)^{n}dx$, find $\lim\limits_{n{\rightarrow}\infty} \frac{a_{n+1}}{a_{n}}$ 4 Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $\int_{x}^{1}f(t)dt\geq(1-x)^{2}$. Prove that $f(1)=0$ 4 Prove that $\int_{0}^{1}f(x)\arctan x dx=\frac{π}{8}\int_{0}^{1}f(x)dx$ 4 Prove that $\lim_{x\to 1^{+} } \int_{x}^{x^{3}}\frac{1}{\ln t}\, dt=\ln3$.

### Reputation (334)

This user has no recent positive reputation changes

 3 Divergence of sequence $y_n=(1-x_1)(1-\ x_2)\cdots(1-x_n)$ if $x_{n+1}=(x_{n+1}+1)x_n$

### Tags (24)

 3 sequences-and-series × 13 0 limits × 11 3 real-analysis × 10 0 calculus × 10 3 convergence × 5 0 limits-without-lhopital × 5 0 integration × 20 0 inequality × 5 0 definite-integrals × 19 0 linear-algebra × 3

### Account (1)

 Mathematics 334 rep 211