Uniform Convergence and limit $(n+1)\int_0^1 x^nf(x) \; dx$
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
111279
The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$
28666
simple convergence test $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$
134637
Exchanging of limits in case of uniform convergence.
111475
Uniform convergence and the supremum theorem
140574
Find $\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx$
1332
Pointwise convergence and convergence of integrals implies $L^1$ convergence
142645
$f_n$ converges uniformly on $[a,b]$ to some function $f$
105971
Convergence of $\sum \limits_{n=1}^{\infty}\sqrt{n^3+1}-\sqrt{n^3-1}$
124382
Showing that $f_n:[0,1]\to\mathbb{R}$ when $f_n=nx(1-x)^n$ converges pointwise but not uniformly
87100
Decide whether $\sum_{n=1}^{\infty}(-1)^n\frac{nx}{1+n^4x^2}$ uniformly converges on $[0,\infty)$ or not
125925
Convergence of $f_n(x)=x^n(1-x)^n$
80440
Limit of $(x_n)$ with $0<x_1<1$ and $x_{n + 1} = x_n - x_n^{n + 1}$
80622
Lebesgue integral uniform convergence
49860
Determine the convergence of $\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{n(n-1)}$
23969
Why isn't $nxe^{-nx}$ uniformly convergent for $x \geq 0$?
103124
Proving that $f(z)=\sum\limits_{n=1}^{\infty}\frac{z}{n(1+n|z|^{2})}$ converges uniformly on $\mathbb{C}$