22 THE SYSTEMS CONCEPT 



Conversion, as described above, is accomplished as follows: 



log A = In A 



2.303 



where 2.303 = log, 10. 



10. Infinite Series,- y = y e ox 



A series is any group of numbers, arithmetically related, which differ from 

 each other in some regular and explicit manner. Thus 



1+2 + 3 + 4 + 5 + n 



is a series. This particular series is divergent, since the larger the n chosen, 

 the greater the sum becomes. There are other series which are convergent, 

 whose value approaches a limit as the number of terms is increased toward 

 infinity. One such convergent series is 



x x 2 x 3 x 4 



1 + — + + '■ + — + 



1 2x1 3x2x1 4x3x2x1 



This series, for a value of x = 1, simplifies to 



1 l 2 l 3 l 4 



1 + — + + + + 



1 2x1 3x2x1 4x3x2x1 



which converges to the numerical value 2.71828 .... as more and more 

 higher index terms are added. In shorthand e x is written for the first, and e x 

 or e for the second series. Thus 



X X L X 3 X* 



and 



e x = 1 + — + — + ■ + ■ + 



1 2x1 3x2x1 4x3x2x1 



1 l 2 l 3 



e = 1 + — + — + + = 2.71828 



1 2x1 3x2x1 



More generally, when x is preceded by a constant, k, kx is substituted for x : 



, , kx (kx) 2 (kxy (kxy 



e** = 1 + h - — — + — + — + 



1 2x1 3x2x1 4x3x2x1 



The constant, k, simply tells how slowly the series converges for any particular 

 value of x : the greater the value of k the greater the number of terms which 

 will be necessary to define e kx to a chosen number of significant figures. 



Now, when x is the variable, and k constant, we can call its evaluation 

 proportional to jv and write 



or y a e kx (1-4) 



