Krdytkyu

1 $begingroup$ Specify whether the series $sum_{n=n_0}^{+ infty } a_n cdot b_n$ must be convergent when: a) $sum a_n$ convergent, $sum b_n$ convergent b) $sum a_n$ convergent absolutely, $sum b_n$ convergent. I need to check my reasoning and I need help to guide them further: If $sum b_n$ convergent, then in particular a sequence of partial sum $sum b_n$ is convergent, so the second condition of Dirichlet's test is met. That is why we should take care of $a_n$ . The first condition of Dirichlet's test is $a_n$ monotonic and $a_n$ convergent to $0$ . So I think in the case of a) the series $sum_{n=n_0}^{+ infty } a_n cdot b_n$ not always is convergent and b) must be convergement, but there are only my thoughts and I do not know how to prove it. r

0 $begingroup$ There is a sequence smooth domain ${Omega_k}$ converges (with uniformly lipschitz constant) to a Lipschhitz domain $Omega$ . Denote by $lambda_k$ and $u_k$ the first eigenvalues and the corresponding frist eigenfunctions of Laplacian in $Omega_k$ . It's known that $lambda_krightarrowlambda$ uniformly as $krightarrowinfty$ , $u_kin C^infty(overline{Omega_k})$ and $uin H_0^1(Omega)$ . Does ${u_k}$ converge to $u$ ? pde share | cite | improve this question asked Dec 20 '18 at 14:12 xiaobiaoJia xiaobiaoJia 18 4