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

$$S = 1 + 1 + 1 + 1 +1 + 1 + \cdots $$ which “sums” to $-\frac{1}{2}$ and

$$S = 1 + 2  + 3 + 4 + 5 + 6 + \cdots $$ which famously “sums” to $-\frac {1}{12}$

Another well known example of a divergent series is the harmonic series. 

$$ S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots$$

The series is so named because every term (besides the first) is the harmonic mean of the two terms next to it thus forming a harmonic progression. These terms also relate to musical harmonics or overtones. We can show the harmonic series is divergent as follows 

\[S = 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right)+ \cdots\]

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.