Ambiguous definition of a sequence in $l^2$ 
Show that the sequence $(x_n)_{n\in\mathbb{N}}$ in $l^2$ given by
  $$
(x_n)_{n\in\mathbb{N}} := \left(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},0,0,0,...\right)
$$
  converges to
  $$x=\left(1, \frac{1}{2},\frac{1}{3},...,\frac{1}{n},\frac{1}{n+1},...\right)$$
  in $l^2$.

Firstly I'm having a problem to understand the first definition. It defines a class of sequences of $l^2$ in the RHS, not just one sequence as stated in the LHS. And to say that this sequence converges to a sequence in $l^2$ it must be the case that it's actually a sequence $((x_n)_{n\in\mathbb{N}})_{n\in\mathbb{N}}$ of sequences of $l^2$. Seems alright?
That being right, I still don't know how to calculate the limit. Given $\varepsilon > 0$, I must find an $N$ such that $n>N$ implies that the sequence $(1,\frac{1}{2},...,\frac{1}{n},0,0,0...)$ is $\varepsilon$-distant to $(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},\frac{1}{n+1},...)$. But $$\|(1,\frac{1}{2},...,\frac{1}{n},0,0,0...)-(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},\frac{1}{n+1},...)\| = \|(0,0,0...,\frac{1}{n+1},\frac{1}{n+2},...)\| = $$
$$
\sum_{k=n+1}^{\infty} \frac{1}{k^2},
$$
and I don't know what to do with this sum. Any clarification will be appreciated.

Your understanding of the first equality is correct: the right-hand side should have been a sequence of elements in $l^2$, instead of a single element of $l^2$. Your notation is incorrect, however: $x_n$ represents a single element of $l^2$ (namely $(1,1/2, \ldots, 1/n, 0, 0, \ldots)$), so the sequence of elements of $l^2$ is $(x_n)_{n \in \mathbb{N}}$. Each $x_n$ is itself a sequence.
For the second part, your work so far is good. Just note that $\sum_{k \ge 1} \frac{1}{k^2}$ is a convergent sequence, so the tail sum $\sum_{k = n+1}^\infty \frac{1}{k^2}$ decreases to zero as $n \to \infty$.
[Finding an explicit expression for $N$ in terms of $\epsilon$ is not really necessary, but if you must, you can bound the tail sum by $\int_n^\infty x^{-2} \, dx$ or something like that.]
