# Real Series

# Why

We want to sum infinitely many real numbers.

# Definition

Let $(a_k)_{k \in \N }$ be a sequence in
$\R $.
Define $(s_n)_{n \in \N }$ by

\[
s_n = \sum_{k = 1}^{n} a_k.
\]

We call $s_n$ the $n$th
partial sum of $(x_k)$.
In other words, the first partial sum $s_1$
is $a_1$, the second partial sum $s_2$ is $a_1
+ a_2$, the third partial sum $s_3$ is $a_1 +
a_2 + a_3$ and so on.
We call $(s_n)$ the sequence
of partial sums or
series of $(a_k)$.
If the series converges,
then we say that $(a_k)$ is
summable.
Clearly not every series is summable: consider,
for example, $a_k = 1$ for all $k$. It has
the divergent series $(1, 2, 3, 4, 5, \dots )$.

## Notation

If the sequence is summable, then there exists
a unique $s \in \R $ (the limit), which we
denote

\[
s = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k = 1}^{n}
a_k.
\]

We read these relations aloud as “s is the
limit as n goes to infinity of s n” and “s
is the limit as n goes to infinity of the sum
of a k from k equals 1 to n.”
We often avoid referencing $s_n$ by abbreviating
the above with
\[
\sum_{k = 1}^{\infty} a_k = s.
\]

We read this notation aloud as “the sum from
1 to infinity of a k is s.”
The notation is subtle, and requires
justification by the algebra of series.
# Convergence

For a series to converge, intuition suggests
that the additional terms added should be
getting smaller and smaller. Indeed:

Let $(a_k)_{k \in \N }$ be a sequence of real
numbers.
If $(a_k)$ is summable then $a_k$ converges to
$0$.

The converse of this theorem has immediate
relevance as a preliminary test for determining
whether a series converges.

If $(a_k)$ does not converge or converges to
$a_0 \neq 0$, then it is not summable.