Having trouble understanding Series and Sequences

So all I could get from my teachers thick accent in class today is that:

A sequence is finite and converges when bounded by x?

and

A series is infinite and diverges because no matter how small the function gets, it will never reach zero?

I'm sorry. I'm really having a hard time understanding the concept. For example. This picture... !

$$\begin{align} s_1 &= a_1\\ s_2 &= a_1+a_2\\ s_3 &= a_1+a_2+a_3\\ s_4 &= a_1+a_2+a_3+a_4\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots\\ s_n &=a_1+a_2+a_3+a_4+\cdots+a_n=\sum_{i=1}^na_i \end{align}$$

I don't the last part involving the sigma. How is that to the statments before it.