Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$?
So do I have to find the values of $p$ where the norm is finite? How do I do this?
$x_n$ converges to $0$ iff $\|x_n\|$ converges to $0$. If $x_n=(x_{n,j})_{j\in N}$ and $1\leq p <\infty $ then $$\|x_n\|=(\sum_{j\in N}|x_{n,j}|^p)^{1/p}.$$ For $1\leq p<\infty $, we have $$\|x_n\|^p=n^2n^{-p}=n^{2-p}$$ so $\|x_n\|=n^{(2/p)-1}$. This converges to $0$ iff $(2/p)-1<0$ iff $2<p$. For $p=\infty$ the def'n of $\|x_n\|$ is $$\|x_n\|=\sup_{j\in N}|x_{n,j}|$$so for $p=\infty$ we have $\|x_n\|=1/n$ which converges to $0$. Altogether we have convergence to $0$ iff $2<p\leq \infty.$