A fine question. Let's take as an example the
power series
(which just means a series over positive (and zero) integer powers of
x, with appropriate coefficients)
P(x) = 1 - x2/3
+ x4/9 -
x6/27 +
x8/81 - ...
Cn is just the
nth
coefficient, so we can list a couple to try and see what's happening:
C0 = 1,
which is just the coefficient of the "0"th power of
x in the
power series. Similarly,
C1,
C2,
C3, and so on are just the
coefficients of the 1st, 2nd and 3rd power of
x, so we have
C1 = 0,
C2 = -1/3,
C3 = 0,
C4 = 1/9
C5 = 0,
C6 = -1/27,
C7 = 0,
C8 = 1/81,
and so on. Can we see a pattern? It looks like the coefficients that
are non-zero alternate sign, and are fractions of successive powers of
3. For
n=2, we have 3
1,
n=4, 3
2, and so on, so we
might guess that the fraction is 1/3
(n/2). Let's check this with
n=6: we get 1/3
6/2 =
1/3
3 = 1/27, which works.
Similarly for
n=8, we get
1/3
8/2 = 1/81. Great!
Now, what about the alternating negative sign? If we take (-1)n, we get +1 for n=0, 2, 4, etc.,
and -1 for n=1, 3, 5, etc. Here we want to alternate between
plus and minus one for n=0, 2, 4, etc. How about
(-1)n/2? Then if n=0,
we get
(-1)0/2 = 1. If n=2,
(-1)2/2 = (-1)1 = -1, and so on, so that works
fine. Putting this together with the fraction we got above, we have
Cn = (-1)n/2 (1/3)n/2.