confused about what 'Series' is? If $(x_n)$ is a sequence then series generated by $(x_n)$ is the sequence $(s_k)$ defined by
$s_1 = x_1$
$s_2 = x_1 + x_2$
$\;\;\;\;\;\vdots$
$s_k = x_1 + x_2 + ... + x_k$
$\;\;\;\;\;\vdots$

Does this mean that series is sequence $s_k$ ?
So series is a special case of sequence?
Thanks.
 A: The most convincing definition of a series might be as a pair of sequences $(x_n)$ and $(s_n)$ related as you explain in your post. Otherwise, a series would be just a sequence $(s_n)$ such that there exists a sequence $(x_n)$ whose $(s_n)$ is the sequence of partial sums. Every sequence $(s_n)$ is such hence the concept of series would become empty, however some texts do adopt this (in my opinion, inferior) convention.
Anyway, a fact to remember is that a series is not uniquely determined by its sum, for example the two series 
$$\sum_{n\geqslant1}\frac1{2^n}\qquad\text{and}\qquad\sum_{n\geqslant1}\frac{41}{(42)^n}
$$ are not equal although they both sum to $1$. One often writes 
$$\sum_{n\geqslant1}x_n\qquad\text{or even}\qquad\sum_{n}x_n
$$ for the series (a sequence or, more rigorously, as explained above, a pair of sequences) and 
$$\sum_{n=1}^\infty x_n
$$ for its sum (a number).
While at it, one might as well mention that a sequence is a function, for example, the real-valued sequence $(x_n)_{n\geqslant1}$ is the function $u:\mathbb N\to\mathbb R$ defined by $u(n)=x_n$ for every $n$ in $\mathbb N$.
A: Each $s_k$ is given by a series of $x_k$. However, the collection $\{s_k\}$ is a sequence.
A series is just a summation of objects belonging to a sequence. You can have a sequence whose elements are individually determined by a series. You can then take a series of a sequence whose elements are individually determined by a series. And so on....
