Whats the difference between a series and sequence? I was looking at a question earlier that involved sequences and found out that the sequence converged to 0 but the series diverged to infinity. How is that possible? for example the sequence was $a_n$ $=$ $\frac{1}{x_n}$ 
The question was here: Convergence of $x_n=f\left(\frac{1}{n}\right), n\geq 1$ 


An essential difference is that
  
  
*
  
*a sequence has  one specific  meaning and
  
*a series has two  different specific meanings.

For convenience only we look at real-valued sequences and series.

Sequences: A sequence of real numbers 
  \begin{align*}
\left(a_n\right)_{n\geq 0}=(a_0,a_1,a_2,\ldots)
\end{align*}
is a function from $\mathbb{N}$ to $\mathbb{R}$. The  index  set  $\mathbb{N}$ is the  domain of the function and    the real   numbers $\mathbb{R}$  is   the codomain. 

Now let's have a look at series. We have to clearly state the meaning(s) of the symbol
\begin{align*}
  \sum_{n=0}^{\infty}a_n\tag{1}
  \end{align*}
with $a_n$ being real numbers.  

Series (as sequences): Infinite series are sequences of partial sums
When looking at a sequence $(a_0,a_{1},a_{2},\ldots)=(a_n)_{n\geq 0}$
  we consider
  \begin{align*}
  (a_0,a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots) = \left(\sum_{n=0}^{N}a_n\right)_{N\geq 0}
  \end{align*}
  and define the infinite series (1) as the sequence of partial sums
  \begin{align*}
    \sum_{n=0}^{\infty}a_n:= \left(\sum_{n=0}^{N}a_n\right)_{N\geq 0}\tag{2}
  \end{align*}

So, the LHS of (2) is just a new and very convenient representation for the sequence of partial sums
\begin{align*}
  \left(s_N\right)_{N\geq 0}, \qquad s_N=\sum_{n=0}^{N}a_n\tag{3}
  \end{align*}
The symbol $\sum_{n=0}^{\infty}a_n$ has also a second meaning, namely

Series (as values): Infinite series are the limits of the sequences of partial sums
An infinite series $\sum_{n=0}^{\infty}a_n$ is said to be convergent, definitely divergent or indefinitely divergent according as the sequence of its partial sums shows the behaviour indicated by those names.
If, in the case of convergence, the sequence of partial sums, $s_N\rightarrow s$, then we say that $s$ is the value or the sum of the convergent infinite series and we write for brevity
\begin{align*}
  \sum_{n=0}^{\infty}a_n:=\lim_{N\rightarrow\infty}s_N=s
  \end{align*}
  so that $\sum_{n=0}^{\infty}a_n$ denotes not only the sequence $(s_N)_{N\geq 0}$ but also the limit $s$, provided this limit exists.

