Let us now look at some divergent series. Again there are two cases:
Case A: the partial sums grow steadily without limit for example
$$S = 1 + 2 + 4 + 8 + 16 + \cdots $$
This is a divergent series and by taking enough terms it can be made to exceed any number no matter how large.
Yet we can do a little bit of “magic” as follows: first multiply the series by two to get
$$2S = 2 + 4 + 8 + 16 + 32 + \cdots$$
subtracting the first equation from the second gives
$$S = -1$$
Thus the series is divergent yet we find it has a “sum” of negative one. The discussion of the meaning and implications of this is outside the scope of this lesson.
Other examples of divergent series that grow steadily without limit are
$$S = 1 + 1 + 1 + 1 + \cdots $$ which “sums” to $-\frac{1}{2}$ and
$$S = 1 + 2  + 3 + 4 + \cdots $$ which famously “sums” to $-\frac {1}{12}$
Another well known example of a divergent series is the harmonic series. 
$$ S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots$$

This series is so named because every term except the first is the harmonic mean of the two adjacent terms. We show the harmonic series to be divergent in two ways below.

$$S = 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right)+ \cdots$$
Next we reduce the right hand side. We reduce $\frac{1}{3}$ to $\frac{1}{4}$. Because 4 is greater than 3 a quarter or 0.25 is less than a third or 0.3333. . . Similarly we reduce $\frac{1}{5}$, $\frac{1}{6}$ and $\frac{1}{7}$ all to $\frac{1}{8}$ to get
$$ S > 1 + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \cdots = 1 + \frac{1}{2} +\frac{1}{2} + \frac{1}{2} + \cdots = \infty$$
So the series is divergent because loosely speaking its sum exceeds infinity.
Another way of showing this is
$$S = \left(1 + \frac{1}{2} + \cdots \frac{1}{9}\right) + \left(\frac{1}{10} + \cdots \frac{1}{99}\right) + \left( \frac{1}{100} + \cdots \frac{1}{999}\right) + \cdots$$
Next we reduce the terms on the right hands side in a different way from what was given above.
$$S > \left(\frac{1}{10} + \cdots \frac{1}{10}\right) + \left( \frac{1}{100} + \cdots \frac{1}{100}\right) + \left( \frac{1}{1000} + \cdots \frac{1}{1000}\right) + \cdots$$
$$S > \left(9 \times \frac{1}{10}\right) + \left( 90 \times \frac{1}{100}\right) + \left(900 \times \frac{1}{1000}\right) + \cdots$$
$$S > \frac{9}{10} + \frac{9}{10} + \frac{9}{10} + \cdots = \infty$$
Case B: the partial sums oscillate forever and never settle down for example
$$S = 1\mathbin{-}1 + 1\mathbin{-} 1 +1\mathbin{-}1 + \cdots$$
The value of this series is 0 if we take an even number of terms and 1 if we take an odd number of terms.
Interestingly we can do what is called Cesaro’s Summation as follows:
$$1\mathbin{-} S = 1\mathbin{-}1 + 1\mathbin{-}1 +1\mathbin{-} \cdots = S $$
which gives $S= \frac{1}{2}$ the average of the two values (1 and 2) we found above. 
Exercises:
1.  The harmonic mean of $a$ and $b$ is defined to be $\frac{2ab}{a +b}$. Show that every term of the harmonic series except the first is the harmonic mean of the two terms next to it.