Summing the terms of a sequence gives a series.

Here is a finite series that ends after five terms:

$$1+2+4+8+16$$

And here is an infinite series with an unlimited number of terms that goes on forever:

$$1+2+4+8+16+\cdots$$

Now consider the infinite series:

$$ a_1 + a_2 + a_3 + a_4 + \cdots + a_n + \cdots$$

We form the following Partial Sums:

$$ S_1 = a_1$$

$$ S_2 = a_1 + a_2$$

$$ S_3 = a_1 + a_2 + a_3 $$

and in general

$$ S_n = a_1 + a_2 + a_3 + \cdots + a_n $$

We thus have the following sequence of partial sums:

$$S_1, S_2, S_3, \cdots, S_n\cdots$$

If this sequence diverges to infinity

$$ \lim_{n\to\infty} S_n = \infty $$ the series is called divergent

On the other hand if the sequence tends to a finite limit 

$$ \lim_{n\to\infty} S_n = S$$ the series is called convergent

in this latter case the remainder after $n$ terms is 

$$ R_n = S \mathbin{-} S_n$$

and

$$\lim_{n\to\infty}R_n=\lim_{n\to\infty}(S\mathbin{-}S_n)=S\mathbin{-}S=0$$