Using the results from pages 361-362, one can write
∑n
i=1
∆Ψ(a+i) =
∑n
i=1
1
a+i= Ψ(a+n+ 1) −Ψ(a+ 1) (12.153)
As an example of how equation (12.153) can be employed, examine the finite sum
S=
1
a+b+
1
a+ 2 b+
1
a+ 3b+···+
1
a+nb (12.154)
This finite series can be expressed
S=^1
b
∑n
i=1
1
a
b+i
=^1
b
∑n
i=1
∆Ψ(a
b
+i) =^1
b
[
Ψ(a
b
+n+ 1) −Ψ(a
b
+ 1)
]
(12.155)
Use differential equations to find series
Another way to find the series representation of a given function is to first
differentiate the function and then form a differential equation satisfied by the given
function. One can then substitute a power series into the differential equation and
determine the coefficients of the power series by comparing like terms as illustrated
in the following examples.
Example 12-9. Determination of series
Find a power series expansion to represent the function y =ax, where ais a
constant.
Solution Write y =y(x) = ax=exln a and differentiate this function to obtain
dy
dx =e
xln aln a=axln a, so that y=axis a solution of the differential equation
dy
dx
=yln a (12.156)
satisfying the initial condition at x= 0,y(0) = a^0 = 1.
Assume that y has the power series representation
y=y(x) = c 0 +c 1 x+c 2 x^2 +c 3 x^3 +···+cnxn+··· (12.157)
with derivative
dy
dx
=y′(x) = c 1 + 2c 2 x+ 3 c 3 x^2 + 4 c 4 x^3 +···+ncnxn−^1 +··· (12.158)
