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 $\displaystyle \sum_{i=1}^\infty a_i$.
| Let
$\displaystyle\displaystyle\sum_{i=1}^\infty a_i$ be a
series, and let $\displaystyle s_n = \sum_{i=1}^n a_i$ be
its $n^{th}$ partial sum. We define $\displaystyle\sum_{i=1}^\infty a_i = \lim_{n \to \infty} s_n.$ If this limit exists and is finite and
equal to $s$, we say the series is convergent
and that $\displaystyle\sum_{i=1}^\infty a_i=s$.
Otherwise, we say the series is divergent.
|