Intereting Posts

Test for an Integer Solution
If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$
Exact sequence arising from symplectic manifold
abelian finite groups – basic
If $G$ has no proper subgroup, then $G$ is cyclic of prime order
Generators of $S_n$
Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$
If A, B, C, D are non-invertible $n \times n$ matrices, is it true that their $2n \times 2n$ block matrix is non-invertible?
Computing limits which involve square roots, such as $\sqrt{n^2+n}-n$
Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?
Finding number of functions from a set to itself such that $f(f(x)) = x$
What does recursive cosine sequence converge to?
The diophantine equation $x^2+y^2=3z^2$
What does it mean for a polynomial to be solvable by radicals?
About finding the period of a function which satisfies $f(x-2)+f(x+2)=f(x)$.

If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$ holds?

I know it holds if we integrate over an interval of the form $[a,b]$ but not sure about improper integral!

- Abel limit theorem
- Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely
- Equicontinuity and uniform convergence 2
- Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?
- Rate of convergence of mean in a central limit theorem setting
- If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

- What is so special about $\alpha=-1$ in the integral of $x^\alpha$?
- Is there another way to solve this integral?
- How to solve the integral $\int \frac {(x^2 +1)}{x^4- x^2 +1} dx$
- Weak Convergence of Positive Part
- Evaluation of $\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$
- Why *all* $\epsilon > 0$, in the $\varepsilon-\delta$ limit definition?
- Simplifying the integral $\int\frac{dx}{(3 + 2\sin x - \cos x)}$ by an easy approach
- Definite integral over a simplex
- How to solve the following integral?
- How find this integral limt

Take $f_n$ a continuous function whose values on $[-n,n]$ is $1/n$, $f_n(x)=0$ if $\left|x\right|\gt n+1$ and $0\leqslant f_n(x)\leqslant 1/n$. Then $\sup_{x\in\mathbf R}\left|f_n(x)\right|=1/n$ from which we deduce that $f_n\to f\equiv 0$ uniformly on the real line. But since $f_n$ is non-negative, $\int_{\mathbf R}f_n(x)\mathrm dx\geqslant \int_0^nf_n(x)\mathrm dx=1$.

In other word, a uniform control is not enough if we integrate over a set of infinite measure.

$$f(x)= \frac{1}{n} \chi_{[n,2n]}$$ will also do.

- Find the expected number of two consecutive 1's in a random binary string
- Showing that $T(n)=2T(+17)+n$ has a solution in $O(n \log n)$
- Topology and axiom of choice
- How to change variables in a surface integral without parametrizing
- What are all of the connected subsets of $\mathbb{Q}$?
- If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
- A sufficient condition for differentiability of a function of several variables in terms of differentiability along paths.
- Finitely generated free modules
- Sampling $Q$ uniformly where $Q^TQ=I$
- Havel & Hakimi degree sequence theory
- Natural logarithms base $e$
- DE solution's uniqueness and convexity
- Why is an integral domain a commutative ring with unity?
- If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$
- Applications of cardinal numbers