310 SAMPLED-DATA CONTROL SYSTEMS 
The technique used in the evaluation of an infinite sum is to express it 
in terms of one of the forms of (11.57), usually the right-hand term. 
Working backwards, the equivalent Laplace transform F(s + jn2r) is 
found and from this F(s). The z transform corresponding to F(s) is 
obtained from the table and stated in terms of e7*, where both T and s are 
set equal to unity. The resultant closed form is that corresponding to 
the left-hand side of (11.57). This is best illustrated by means of an 
example. 
EXAMPLE 
It is desired to evaluate the following infinite series: 
+ 
1 
s= ) are 
n=— 0 
The general term in the summation is expressed as 
1 1 
n? + a? oe (a + jn)(a — jn) 
which is identical to 

Ar? 
(27a + j2rn)(2ra — j2rn) 

This, in turn, is equal to 
Aq? 
[2ra + s + (920/T)n\[2ra — s — (720/T)n] \n=0 

The summation of terms of the latter type constitutes the right-hand 
side of (11.56). This latter expression stems from a Laplace transform 
F(s), given by 
Arr? 
ES) Fon ara (ea ae) 
which can be expanded into partial fractions 
ea ee a 
F(s) SG: eer 
The z transform corresponding to F(s), setting 7 to unity and s to zero 
after replacing z by e”*, is 
3) eI I a ea Eo oe 
Ce a ee ee pte 
This is the value of the summation of the form on the left-hand side of 
(11.57), which is equal to the desired summation. 
F(e?*) 

