About Divergent Series Are all divergent series equal each other?
As an example, can we say that $\sum_{k=1}^{\infty}k^2$ and $\sum_{k=1}^{\infty}k^3$ are equal?
 A: no consider the subtraction $\sum_{k=1}^{\infty}k^3-k^2$ this is still divergent therefore not all divergent series are equal. (if all divergent sums were equal then the subtraction would converge to 0)
A: A series as a mathematical object really is a sequence (the sequence of partial sums). Therefore to say two series are equal means that their partial sums are equal (even further their terms are equal). Therefore two series are only equal to each other if they are the exact same series.
Now, what you were really referring to in the question was the limit of a series (you might call it the value of the series). Many people tend to identify a series with its value in their mind. In this light, it still would not be good to call two divergent series equal because they have no values to compare.
I would caution against identifying a series with its value. While the value of a series is an important characteristic of it, it isn't the only thing worth knowing about the series. For instance, one important and interesting thing to know is how "fast" the series converges.
For instance, $2 + \sum_{k=1}^\infty 0$ and $\sum_{k=0}^\infty (\frac{1}{2})^k$ both converge to $2$, but the former converges much faster than the latter and they are very different by nature.
A: A series $\sum_{n=1}^{\infty}a_n$ is a pair of sequences, the sequence $(a_n)$ of terms, and the sequence $(S_n)$ of partial sums, where $S_n = \sum_{k=1}^{n} a_k.$ When we write $\sum_{k=1}^{\infty}a_n = S \in \mathbb R,$ we mean $\lim_{n\to \infty} S_n = S,$ and say the series converges to $S,$ or that the sum of the series is $S.$ We can allow $\pm \infty$ into the game as possible values of $S,$ although in those cases we would say the series diverges to $S.$
Now when you see $\sum a_n = \sum b_n,$ it usually means the two sums are the same, but not that the series are the same. So writing $\sum k^2 = \sum k^3$ makes perfect sense in that regard. I myself would not say the series are the same, but that their sums are. 
