Is $(1,0,...,0,-n^2,0,0,...) \in \ell^2$?

Question

I am a bit unsure of this because if $n$ is very large then the sum would not be finite but then again the term after the nth term is $0$ onwards.

Answer 1

An infinite-dimensional vector $x = (x_1, x_2, x_3, \ldots)$ (or I guess we can just say sequence) is in $\ell^2$ iff $\displaystyle \sum_{k=1}^{+\infty} |x_k|^2 < +\infty$. Well, no matter what the value of $n$ is in $x = (1, 0, 0, \dots, 0, -n^2, 0, 0, 0, \dots)$, we'll always have $\displaystyle \sum_{k=1}^{+\infty} |x_k|^2 = 1 + n^4$, and this is clearly finite for every integer (or real, etc.) value of $n$.

So, yes, the sequence is in $\ell^2$.