## Mean Partial Sums of Non-Convergent Series

```If the partial sums of an infinite series converge on a finite
value the series is said to be convergent, whereas if the partial
sums increase (in magnitude) without limit, the series is said to
be divergent.  In addition to these two kinds of series, there is
another category of series - which may be called non-convergent -
whose partial sums are bounded in magnitude and yet do not converge
on any finite value.  For example, for any real number x, consider
the infinite series

sin(0x) + sin(1x) + sin(2x) + sin(3x) + ...

The individual terms are each in the range -1 to +1, and they
eventually precess around the entire cycle (modulo 2pi), resulting
in an upper bound on the partial sums.  Interestingly, the mean value
of all the partial sums converges to give the interesting identity

1   m    k                 1      / x - pi \
F(x)  =   lim    --- SUM  SUM  sin(jx) =  - --- tan( -------- )
m->inf   m  k=0  j=0                2      \   2    /

Thus the mean value of the partial sums of sines with uniformly
distributed arguments is simply a re-scaled and shifted version of
tangent.  This function can also be written in the form

1 + cos(x)
F(x)  =   ------------
2 sin(x)

It's interesting to consider the continuous analog of F(x), which
we may define by replacing the summations with integrations as
shown below
m    k
1   /    /                      1
f(x)  =   lim    ---  |    |  sin(jx) dj dk   =  ---
m->inf   m   /    /                      x
k=0  j=0

Thus the function f(x) based on continuous integrals goes to
zero as x increases, whereas the function F(x) based on discrete
summations is periodic.  These two functions do, however, approach
each other for small values of x, as shown by the power series
expansion of F(x)

1       1         1
F(x)  =  ---  -  --- x  -  --- x^3 - ...
x       12       720

A similar construction based on the cosine instead of the sine is
not as interesting, because if we consider the sum

cos(0x) + cos(1x) + cos(2x) + ...

with x equal to 2n pi, the terms are all +1, and so the partial sums
are unbounded.  In between these divergent points the limiting
function is simply the constant 1/2, basically because the cosine
is an even function.  By the way, it's amusing to note that the
singularities at 2n pi are removable, so in this sense we can actually
justify Euler's bold claim that the sum of 1-1+1-1+1-... is 1/2, as
suggested by the geometric series 1 + x + x^2 + x^3 +... = 1/(1-x)
with x=-1 (where the series is non-convergent).  The difference is
that here we have said the average of the partial sums can be
analytically continued as 1/2 at x=2pi, rather than suggesting the
partial sums themselves converge.  Of course, by taking the average
of partial sums we impose a particular order (arrangement) on the
terms.  For a related discussion, see Formal-Numeric Series.
```