Section 5.2 Infinite Series
Given a sequence \(\ds\{a_n\}_{n=1}^\infty\) we can easily sum the first \(N\) terms, giving
\begin{equation}
S_N := a_1 + a_2 + \cdots + a_N = \sum_{n=1}^N a_n\tag{5.2.1}
\end{equation}
(Note that we have to use a dummy index \(n\) to indicate each term in the sum that is different from the index \(N\) indicating the number of terms in that sum.)
This is analogous to a definite integral \(\int_1^N f(x)\, dx.\)
It can also be useful to make sense of summing all the terms of an infinite sequence, to get
\begin{equation*}
S = a_1 + a_2 + \cdots + a_n + \cdots
\end{equation*}
which is analogous to an improper integral \(\int_1^\infty f(x)\, dx\text{.}\) To do this, we use the same strategy as with improper integrals:
Definition 5.2.1. Infinite Sum, or Series.
The infinite sum of sequence \(\ds \{a_n\}_{n=1}^\infty\) is
\begin{equation}
\sum_{n=1}^\infty a_n = \lim_{N \to \infty} S_N = \lim_{N \to \infty} \sum_{n=1}^N a_n,\tag{5.2.2}
\end{equation}
if this limit exists with a finite value, and we then say that the series (infinite sum) converges.
If there is not a finite limit, we say that the series (infinite sum) diverges.
If there is an infinite limit, we sometimes say that the series (infinite sum) diverges to infinity.
The finite sum \(S_N\) up to term \(N\) of the sequence is called the \(N\)-th partial sum of the series.
Checkpoint 5.2.2.
For the sequence \(\ds \left\{ \frac{1}{2^{n-1}} \right\}_{n=1}^\infty = \left\{ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots \right\}\)
Show that the \(N\)-th partial sum \(\ds S_N = \sum_{n=1}^N \frac{1}{2^n} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^{N-1}} = 2- \frac{1}{2^{N-1}}\) (Aside: the sequence of partial sums is thus bounded above.)
Show that the series (infinite sum) converges, with value \(\ds \sum_{n=1}^\infty \frac{1}{2^n} = 2\text{.}\)
Geometric sequences and series.
The above example is one case of a geometric series, one of the few situations where we can get an easily evaluated expression for the sum of an infinite sequence. A geometric sequence is one in which the ratio of every pair of consecutive terms has the same value \(r\text{.}\) Thus calling the first term \(a\text{,}\) the remaining terms are given recursively by \(a_n = r a_{n-1}\) and this leads to the explicit form
\begin{equation*}
a_n = ar^{n-1}, n \geq 1.
\end{equation*}
For \(r \neq 1\text{,}\) the partial sums can be shown to be \(\ds S_N = a\frac{1-r^N}{1-r}\text{.}\) Then for \(|r| < 1\text{,}\) these partial sums have a limit, giving this geometric series the value
\begin{equation*}
\sum_{n=1}^\infty ar^{n-1} = a +ar + ar^2 + \cdots + ar^{n-1} + \cdots = \frac{a}{1-r}, \; |r| < 1.
\end{equation*}
Geometric series: divergence for \(|r|\geq 1\).
What happens if instead \(|r| \geq 1\text{?}\) The sum diverges, but in a variety of different ways:
For \(r>1\text{,}\) the partial sums grow without bound and have no [finite] limit: the series diverges to an infinite value; for example, with \(a=1\text{,}\) \(r=2\) gives the sum form \(1+2+4+8+ \cdots\) so the partial sums are going towards \(\infty\text{,}\) and the series diverges to infinity.
For \(r < -1\text{,}\) the partial sums oscillate ever more wildly, and diverge without even an infinite limit; with \(a=1\text{,}\) \(r=-2\) gives the sum form \(1-2+4-8+ \cdots\) so the partial sums do not even have an infinite limit.
For \(r=1\text{,}\) \(a_n=a\) so \(S_N = a N\text{,}\) and the series again diverges to infinity (except in the trivial case \(a=0\text{,}\) where \(S_N=0\text{.}\))
Finally, for \(r=-1\text{,}\) the sequence is \(\{a, -a, a, -a, \dots \}\text{,}\) so the successive partial sums are \(a, 0, a, 0, \dots\text{;}\) again not converging to any value (except if \(a=0\)).
Other starting points for sums.
It is useful to define infinite sums for other starting points too:
\begin{equation*}
\sum_{n=b}^\infty a_n = \lim_{N \to \infty} \sum_{n=b}^N a_n.
\end{equation*}
For example, a geometric sequence is more elegantly described with indices starting at 0:
\begin{equation*}
\{ ar^n \}_{n=0}^\infty = \{ ar^0, ar^1, ar^2, \dots \} = \{ a, ar, ar^2, \dots \}
\end{equation*}
and the most elegant way to state the result for convergent geometric series is
\begin{equation*}
\sum_{n=0}^\infty ar^n = \frac{a}{1-r}, \; |r| < 1.
\end{equation*}
The Harmonic Series: the terms converge to zero, but the series diverges!
The harmonic series is
\begin{equation*}
\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac12 + \frac 13 + \cdots
\end{equation*}
and is unusual in that it is important enough to have a name, yet it does not have a value, because it diverges [to infinity]. It is also an important cautionary example: our first example where the sequence of terms has limit zero [\(1/n \to 0\)], but the infinite sum of these terms diverges. One way to see that the sum diverges is to group terms like this:
\begin{equation*}
1 + \frac12 + \left( \frac 13 + \frac14 \right) + \left( \frac15 + \frac16 + \frac17 + \frac18 \right) + \cdots
\end{equation*}
Each term in parentheses has value of at least
\(1/2\text{,}\) so the partial sums are at least
\(1/2 + 1/2 + 1/2 + \cdots\) and increase to arbitrarily large values. We will see another way to show this divergence in
Section 5.3 Terms going to zero is necessary but not sufficient for series convergence.
There seems to be some connection between the terms of a sequence going to zero and its infinite sum converging, though the harmonic series shows that terms going to zero in not always enough to guarantee convergence of the infinite sum.
There is a guarantee in the other direction though:
Theorem 5.2.3.
If a series \(\sum a_n\) converges, then its terms must converge to zero: \(a_n \to 0\text{,}\) but not conversely!
Rules for sums and constant multiples.
Series have some familiar properties, akin to rules for integrals, limits of sequences, and derivatives:
\begin{equation*}
\begin{split}
\sum_{n=1}^\infty c a_n \amp = c \sum_{n=1}^\infty a_n
\\
\sum_{n=1}^\infty (a_n + b_n) \amp = \left( \sum_{n=1}^\infty a_n \right) + \left( \sum_{n=1}^\infty b_n \right)
\\
\sum_{n=1}^\infty (a_n - b_n) \amp = \left( \sum_{n=1}^\infty a_n \right) - \left( \sum_{n=1}^\infty b_n \right)
\end{split}
\end{equation*}
and of course the sum can instead start at any value on \(n\text{,}\) not just at \(n=1\text{.}\)
The first few terms do not matter for convergence/divergence.
Just as with limits of sequences, whether a series converges does not depend on the first few terms—where by “few”, I mean any finite number of them!
If you add any number of extra terms at the start of a series, or remove any number of terms, or change the values of any finite number of the terms, that can only change the value of the series (sum) by a finite amount; it cannot convert a convergent series into a divergent one, or vice versa.
Study Guide.
The Definition in terms of Partial Sums and Limits
The Definition and properties of Geometric Series
Theorem 7, which shows that series have a lot of properties in common with improper integrals
Examples 7 [parts (a) and (b)], 8, 9
Checkpoints 7, 8, 9
and one or several exercises from each of the following groups: 67–70, 71–74, 79 & 80, 83–86, 87–92, 93–96.
openstax.org/books/calculus-volume-2/pages/5-2-infinite-series
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