Infinite series are essentially sequences formed by summing the terms of another sequence. In this section we define what infinite series are, and what it means for them to converge.
Definition
Let (a_n) be a sequence of real numbers. A partial sum of (a_n) is given by
a_{m} + a_{m+1} + \cdots a_{n}
where m \leq n.
Partial sums are a partial collection of terms of the sequence added together.
Let (a_n) be a sequence of real numbers. We notate the partial sum from m to n by
\sum^n_{k=m} a_k
We may also save time by letting
\sum a_n := \sum^{\infty}_{n=0} a_n \text{ or } \sum^{\infty}_{n=1} a_n
where are index starts at 0 or 1, whichever makes sense in context.
Let (a_n) be a sequence of real numbers. We define a sequence (s_n) of partial sums by
s_n = \sum^n_{k=1} a_k = a_1 + a_2 + \cdots + a_n.
We refer to the sequence (s_n) as an infinite series.
We define (s_n) as a sequence of partial sums in order to formalize the series into a sequence, which we’ve already studied. All of our previous theorems will apply to (s_n), and therefore we can prove many theorems essentially for free!
Convergence
If (s_n) converges to a real number s, we say the series \sum a_n is convergent and write
\sum^\infty_{n=1}a_n = s.
If (s_n) does not converge, the series \sum^\infty_{n=1}a_n is divergent. If
\begin{align*}
\lim_{n \to \infty} s_n &= +\infty \quad \text{then} \quad \sum^\infty_{n=1}a_n = +\infty \quad \text{ and}\\
\lim_{n \to \infty} s_n &= -\infty \quad \text{then} \quad \sum^\infty_{n=1}a_n = -\infty.
\end{align*}
The question of whether or not a series converges is of great interest to mathematicians.
The harmonic series is defined
\sum^\infty_{n=1} \frac{1}{n}
The sequence (s_n) is divergent, and thus the harmonic series is divergent. This is
The series
\sum^\infty_{n=1} \frac{1}{n(n+1)}
has the partial sum
s_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \cdots + \frac{1}{n(n+1)}
Let \sum a_n = s and \sum b_n = t. Then
(a) \sum (a_n + b_n) = s + t and
(b) \sum (ka_n) = ks for every k \in \mathbb{R}.
(a) By converting our series to limits, we obtain
\sum (a_n + b_n) = \lim (s_{a_n} + s_{b_n}) = s + t.
(b) Similarly, we obtain
\sum (ka_n) = \lim ks_{a_n} = k\lim s_{a_n} = ks.
If \sum a_n is a convergent series, then \lim_{n \to \infty} a_n = 0.
If \sum a_n is a convergent series, then the sequence of partial sums (s_n) must have a finite limit.
Since a_n = s_n - s_{n-1}, we have \lim a_n = \lim s_n - \lim s_{n-1} = 0.
The infinite series \sum a_n converges if and only if for each \varepsilon > 0, there exists a natural number N such that if n \geq m \geq N, then
|a_m + a_{m+1} + \cdots + a_{n}| = |s_n - s_{m-1}| < \varepsilon.
Let \sum a_n be a convergent series. Then the sequence (s_n) of partial sums converges. By the Cauchy Convergent Criterion, (s_n) is Cauchy. Thus, for any \varepsilon > 0, there exists N \in \mathbb{N} such that m,n \geq N implies |s_n - s_m| < \varepsilon. Hence, if n \geq m \geq N + 1, then m - 1 \geq N, and |s_n - s_{m-1}| < \varepsilon as desired.
Conversly, for all \varepsilon > 0 let N exists such that n \geq m \geq N implies |s_n - s_{m-1}| < \varepsilon. Then for n > m \geq N we have m + 1 > N, so that |s_n - s_m| < \varepsilon. This implies (s_n) is Cauchy, and therefore convergent.
Recap
Let \sum a_n = s and \sum b_n = t. Then
(a) \sum (a_n + b_n) = s + t and
(b) \sum (ka_n) = ks for every k \in \mathbb{R}.
If \sum a_n is a convergent series, then \lim_{n \to \infty} a_n = 0.
The infinite series \sum a_n converges if and only if for each \varepsilon > 0, there exists a natural number N such that if n \geq m \geq N, then
|a_m + a_{m+1} + \cdots + a_{n}| = |s_n - s_{m-1}| < \varepsilon.