Consider $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$. We know that it converges.
Given $k\in\left(0,\infty\right)$. Is it then "okay" to say that there exists a $j\in\mathbb{N}$ such that $\sum_{n=j}^{\infty} \frac{1}{n^{3}}
Consider $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$. We know that it converges.
Given $k\in\left(0,\infty\right)$. Is it then "okay" to say that there exists a $j\in\mathbb{N}$ such that $\sum_{n=j}^{\infty} \frac{1}{n^{3}}
It always depends on the audience, but for an audience knowledgeable about sequences and series this should be obvious enough not to require further detailed arguments.
The fact $\sum\limits_{n=1}^{\infty} \frac{1}{n^{3}}$ that converges means that converges the sequence of partial sums $S_N=\sum\limits _{n=1}^{N} \frac{1}{n^{3}}$