Difference between a series and a sequence of partial sums I understand what a series is. 
I understand what a partial sum of a series is.
But what is a sequence of partial sums? As i understand it, it's the same thing that a series.
The expression: $1+2+3+4+5+\cdots$ is a series.

The sequence of partial sums of that series is: $1, 3, 6, 10, 15, \ldots$.

Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.

Remember: a series is a sum; a sequence is a list.
A series $\sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{n\geq n_0}$ is the sequence of partial sums given by

$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$