Intereting Posts

Find the number of all subsets of $\{1, 2, \ldots,2015\}$ with $n$ elements such that the sum of the elements in the subset is divisible by 5
Why infinity multiplied by zero was considered zero here?!
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Elements as a product of unit and power of element in UFD
The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)
$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$
Definition of a Group in Abstract Algebra Texts
Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$
Not divisible by $2,3$ or $5$ but divisible by $7$
Closed form for an infinite series of Bessel functions

**Assume that $f$ is twice diffentiable on $R$,and such**

$$2f(x)+f”(x)=-xf'(x)$$

**show that:**

- $\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$
- Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
- Geometric series of an operator
- liminf in terms of the point-to-set distance
- Showing a function is not uniformly continuous
- Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

$f(x)$and $f'(x)$ are bounded on $R$

My try:since

$$2f(x)+f”(x)+xf'(x)=0$$

and following I can’t any work,Thank you very much!

- Are rotations of $(0,1)$ by $n \arccos(\frac{1}{3})$ dense in the unit circle?
- Approximation by smooth function while preserving the zero set
- Points of discontinuity of a bijective function $f:\mathbb{R} \to [0,\infty)$
- Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?
- Dini's Theorem and tests for uniform convergence
- Understanding a Proof for Why $\ell^2$ is Complete
- Characterising Continuous functions
- Is there a non-compact metric space, every open cover of which has a Lebesgue number?
- Why is the rational number system inadequate for analysis?
- Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Define

$$g=f^2+\frac{1}{2}(f’)^2.$$

By definition, $g$ is non-negative and differentiable; moreover,

$$g'(x)=f'(x)\cdot\left(2f(x)+f”(x)\right)=-x\cdot (f'(x))^2,\quad\forall x\in \Bbb R.$$

Therefore, $g$ is increasing on $(-\infty,0]$ and decreasing on $[0,+\infty)$, so $g(\Bbb R)\subset [0, g(0)]$. The conclusion follows.

- Definition of a monoid: clarification needed
- Prove $0$ is a partial limit of $a_n$
- Lifting idempotents modulo a nilpotent ideal
- Is this space contractible?
- Find all entire $f$ such that $f(f(z))=z$.
- Show that a nonabelian group must have at least five distinct elements
- Are there infinitely many Mersenne primes?
- find partial derivative of $a_1x+b_1y$ with respect to $a_2x+b_2y$
- Inequality involving inradius, exradii, sides, area, semiperimeter
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- Convergence of $\int_{0}^{+\infty }\frac{e^{-x}} {1+x^2}dx$
- find maximum area
- Spectral radius inequality
- Zero-dimensional ideals and finite-dimensional algebras