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.