While much more can be said about sequences, we now turn to our
principal interest, series. Recall that a series, roughly speaking, is
the sum of a sequence: if $\ds\{a_n\}_{n=0}^\infty$ is a sequence then the
associated series is
$$\sum_{i=0}^\infty a_n=a_0+a_1+a_2+\cdots$$
Associated with a series is a second sequence, called the
**sequence of partial sums**
$\ds\{s_n\}_{n=0}^\infty$:
$$s_n=\sum_{i=0}^n a_i.$$
So
$$s_0=a_0,\quad s_1=a_0+a_1,\quad s_2=a_0+a_1+a_2,\quad \ldots$$
A series converges
if the sequence of partial sums converges, and otherwise the series
diverges.

Example 13.2.1
If $\ds a_n=kx^n$, $\ds\sum_{n=0}^\infty a_n$ is called a
**geometric series**.
A typical partial sum is
$$s_n=k+kx+kx^2+kx^3+\cdots+kx^n=k(1+x+x^2+x^3+\cdots+x^n).$$
We note that
$$\eqalign{
s_n(1-x)&=k(1+x+x^2+x^3+\cdots+x^n)(1-x)\cr
&=k(1+x+x^2+x^3+\cdots+x^n)1-k(1+x+x^2+x^3+\cdots+x^{n-1}+x^n)x\cr
&=k(1+x+x^2+x^3+\cdots+x^n-x-x^2-x^3-\cdots-x^n-x^{n+1})\cr
&=k(1-x^{n+1})\cr
}$$
so
$$\eqalign{
s_n(1-x)&=k(1-x^{n+1})\cr
s_n&=k{1-x^{n+1}\over 1-x}.\cr
}$$
If $|x|< 1$, $\ds\lim_{n\to\infty}x^n=0$ so
$$
\lim_{n\to\infty}s_n=\lim_{n\to\infty}k{1-x^{n+1}\over 1-x}=
k{1\over 1-x}.
$$
Thus, when $|x|< 1$ the geometric series converges to $k/(1-x)$. When, for
example, $k=1$ and $x=1/2$:
$$
s_n={1-(1/2)^{n+1}\over 1-1/2}={2^{n+1}-1\over 2^n}=2-{1\over 2^n}
\quad\hbox{and}\quad \sum_{n=0}^\infty {1\over 2^n} =
{1\over 1-1/2} = 2.
$$
We began the chapter with the series
$$\sum_{n=1}^\infty {1\over 2^n},$$
namely, the geometric series without the first term $1$. Each partial
sum of this series is 1 less than the corresponding partial sum for
the geometric series, so of course the limit is also one less than the
value of the geometric series, that is,
$$\sum_{n=1}^\infty {1\over 2^n}=1.$$
$\square$

It is not hard to see that the following theorem follows from theorem 13.1.2.

Theorem 13.2.2 Suppose that $\sum a_n$ and $\sum b_n$ are convergent series, and $c$ is a constant. Then

1. $\ds\sum ca_n$ is convergent and $\ds\sum ca_n=c\sum a_n$

2. $\ds\sum (a_n+b_n)$ is convergent and $\ds\sum (a_n+b_n)=\sum a_n+\sum b_n$.

$\qed$

The converses of the two parts of this theorem are subtly different. Suppose that $\sum a_n$ diverges; does $\sum ca_n$ also diverge if $c$ is non-zero? Yes: suppose instead that $\sum ca_n$ converges; then by the theorem, $\sum (1/c)ca_n$ converges, but this is the same as $\sum a_n$, which by assumption diverges. Hence $\sum ca_n$ also diverges. Note that we are applying the theorem with $a_n$ replaced by $ca_n$ and $c$ replaced by $(1/c)$.

Now suppose that $\sum a_n$ and $\sum b_n$ diverge; does $\sum (a_n+b_n)$ also diverge? Now the answer is no: Let $a_n=1$ and $b_n=-1$, so certainly $\sum a_n$ and $\sum b_n$ diverge. But $\sum (a_n+b_n)=\sum(1+-1)=\sum 0 = 0$. Of course, sometimes $\sum (a_n+b_n)$ will also diverge, for example, if $a_n=b_n=1$, then $\sum (a_n+b_n)=\sum(1+1)=\sum 2$ diverges.

In general, the sequence of partial sums $\ds s_n$ is harder to understand and analyze than the sequence of terms $\ds a_n$, and it is difficult to determine whether series converge and if so to what. Sometimes things are relatively simple, starting with the following.

**Proof.**
Since $\sum a_n$ converges, $\ds\lim_{n\to\infty}s_n=L$ and
$\ds\lim_{n\to\infty}s_{n-1}=L$, because this really says the same
thing but "renumbers'' the terms. By
theorem 13.1.2,
$$
\lim_{n\to\infty} (s_{n}-s_{n-1})=
\lim_{n\to\infty} s_{n}-\lim_{n\to\infty}s_{n-1}=L-L=0.
$$
But
$$
s_{n}-s_{n-1}=(a_0+a_1+a_2+\cdots+a_n)-(a_0+a_1+a_2+\cdots+a_{n-1})
=a_n,
$$
so as desired $\ds\lim_{n\to\infty}a_n=0$.
$\qed$

This theorem presents an easy divergence
test: if given a series $\sum
a_n$ the limit $\ds\lim_{n\to\infty}a_n$ does not exist or has a value
other than zero, the series diverges. Note well that the converse is
*not* true: If $\ds\lim_{n\to\infty}a_n=0$ then the series does
not necessarily converge.

Example 13.2.4 Show that $\ds\sum_{n=1}^\infty {n\over n+1}$ diverges.

We compute the limit: $$\lim _{n\to\infty}{n\over n+1}=1\not=0.$$ Looking at the first few terms perhaps makes it clear that the series has no chance of converging: $${1\over2}+{2\over3}+{3\over4}+{4\over5}+\cdots$$ will just get larger and larger; indeed, after a bit longer the series starts to look very much like $\cdots+1+1+1+1+\cdots$, and of course if we add up enough 1's we can make the sum as large as we desire. $\square$

Example 13.2.5 Show that $\ds\sum_{n=1}^\infty {1\over n}$ diverges.

Here the theorem does not apply: $\ds\lim _{n\to\infty} 1/n=0$, so it looks like perhaps the series converges. Indeed, if you have the fortitude (or the software) to add up the first 1000 terms you will find that $$\sum_{n=1}^{1000} {1\over n}\approx 7.49,$$ so it might be reasonable to speculate that the series converges to something in the neighborhood of 10. But in fact the partial sums do go to infinity; they just get big very, very slowly. Consider the following:

$\ds 1+{1\over 2}+{1\over 3}+{1\over 4} > 1+{1\over 2}+{1\over 4}+{1\over 4} = 1+{1\over 2}+{1\over 2}$

$\ds 1+{1\over 2}+{1\over 3}+{1\over 4}+ {1\over 5}+{1\over 6}+{1\over 7}+{1\over 8} > 1+{1\over 2}+{1\over 4}+{1\over 4}+{1\over 8}+{1\over 8}+{1\over 8}+{1\over 8} = 1+{1\over 2}+{1\over 2}+{1\over 2}$

$\ds 1+{1\over 2}+{1\over 3}+\cdots+{1\over16}> 1+{1\over 2}+{1\over 4}+{1\over 4}+{1\over 8}+\cdots+{1\over 8}+{1\over16}+\cdots +{1\over16} =1+{1\over 2}+{1\over 2}+{1\over 2}+{1\over 2}$

and so on. By swallowing up more and more terms we can always manage
to add at least another $1/2$ to the sum, and by adding enough of
these we can make the partial sums as big as we like. In fact, it's
not hard to see from this pattern that
$$1+{1\over 2}+{1\over 3}+\cdots+{1\over 2^n} > 1+{n\over 2},$$
so to make sure the sum is over 100, for example, we'd add
up terms until we get to around $\ds 1/2^{198}$, that is,
about $\ds 4\cdot 10^{59}$ terms. This series, $\sum (1/n)$, is called the
**harmonic series**.
$\square$

## Exercises 13.2

**Ex 13.2.1**
Explain why $\ds\sum_{n=1}^\infty {n^2\over 2n^2+1}$
diverges.
(answer)

**Ex 13.2.2**
Explain why $\ds\sum_{n=1}^\infty {5\over 2^{1/n}+14}$
diverges.
(answer)

**Ex 13.2.3**
Explain why $\ds\sum_{n=1}^\infty {3\over n}$
diverges.
(answer)

**Ex 13.2.4**
Compute $\ds\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}$.
(answer)

**Ex 13.2.5**
Compute $\ds\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}$.
(answer)

**Ex 13.2.6**
Compute $\ds\sum_{n=0}^\infty {4^{n+1}\over 5^n}$.
(answer)

**Ex 13.2.7**
Compute $\ds\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}$.
(answer)

**Ex 13.2.8**
Compute $\ds\sum_{n=1}^\infty \left({3\over 5}\right)^n$.
(answer)

**Ex 13.2.9**
Compute $\ds\sum_{n=1}^\infty {3^n\over 5^{n+1}}$.
(answer)