# Tagged Questions

Convergence of sequences and different modes of convergence.

8 views

### Heat Ecuation : Exchange partial derviate and series

I'am having a problem when checking the validity of the solution i found for the heat ecuation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
19 views

### Integral Convergence with parameters

I am finding it hard to approach this question: $$\int_0^{\pi/2} {1-\cos(x)^a\over x^b}\, dx$$ and I need to determine for which positive values of $a,b$ the integral converges. Thanks,
36 views

44 views

49 views

28 views

### Pointwise convergence: State $f(x) = \lim f_n(x)$

I'm completely confused by this subject and hoping you guys can help me to clear up my confusion. So I'm told: State $f(x) = \lim f_n(x)$ where $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ ...
199 views

### If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed using a “direct” proof

Rudin Exercise 4.25(a) reads: If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed. The hints in the problem suggest a proof by proving that the complement of $K + C$ ...
57 views

### When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
286 views

### Prove $\cos(n)$ does not converge as $n$ tends to infinity

How do I go about proving that $\lim\limits_{n \to \infty} \cos(n)$ does not exist where $n\in \mathbb{N}$ using an $\epsilon-N$ style method?
### On the convergence of a series $\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n$
I am struggling with the series $$\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n,$$ where $p,q>0$. I have checked the Dirichlet and ratio test so far, ...