Introduction to Series

An series is an infinite sum, which we think of as the sum of the terms of a sequence, $a_1 + a_2 + a_3 + \ldots.$  We write a series using summation notation as $$\sum_{n=1}^\infty a_n=a_1 + a_2 + a_3 + \cdots.$$DO:  Convince yourself that  $\displaystyle\sum_{i=1}^\infty a_i=\sum_{k=1}^\infty a_k=\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\cdots $.  The $i,k$, and $n$ are just indices.
Notice: 
a sequence is a list of infinitely many terms $a_1,a_2,a_2\ldots$, but 
a series is a sum of infinitely many terms $a_1 + a_2 + a_3 + \cdots$.

Addition is something we do to finitely many numbers, not to infinitely many - how can we add infinitely many numbers?  Consider a series $\displaystyle\sum_{i=1}^\infty a_i=a_1 + a_2 + a_3 + \cdots.$  Instead of worrying about adding infinitely many $a_i$, we take (finite) partial sums $s_n$, where $s_n$ is the sum of the first $n$ terms of the series.$$\begin{eqnarray*}s_1&=&a_1\\s_2&=&a_1+a_2\\s_3&=&a_1+a_2+a_3\\&\vdots&\\s_n&=&a_1+a_2+\cdots+a_{n-1}+a_n\\   \end{eqnarray*}$$A partial sum makes perfect sense as a sum of finitely many numbers.  We can, of course, write each of these partial sums in summation notation; for example  $$ s_2=\sum_{i=1}^2a_i, \qquad s_5=\sum_{i=1}^5 a_i,\qquad s_n=\sum_{i=1}^n a_i, \quad\text{ etc.}$$Notice that as $n$ gets larger and larger, $\displaystyle s_n=\sum_{i=1}^n=a_1+a_2+\cdots+a_n$ gets closer and closer to the infinite sum $\displaystyle \sum_{i=1}^\infty a_i$.