Undetermined Coefficients. 159 



Query : What controls the law of the series in the develop- 

 ment of a fraction ? 



Query : How does the numerator affect the development of a 

 fraction in the form of a series ? 



Query : What would the results to examples 7 and 8 suggest 

 about the development of fractions which are reciprocals ? 



10. It sometimes happens when we try to develope a fraction 

 by the method explained that some of the equations are absurd 

 or contradict one another. 



The reason of this is because the fraction cannot be developed 

 into a series of the form assiuned. Thus if we tr>' to develope 



I 



we assume 



x—x- 



X — X- 12 3 4 



Multiply by x—x" and we get 



i==AXA AJx'-^-f rA,AJ.r^-f ... 

 hence 1=0, 



A=o, 



A=o, 



A^=-o, 



etc. 



But the first of these equations is false, so we consider that 



the function cannot be developed into a series of the assumed 



form. 



But we note the denominator of the given fraction contains a 



factor .V, and that hence the fraction proposed equals • , the 



second factor of which has already been developed. 



From this we would infer that the development of „ could 



be obtained from the development of by dividing every term 



in that development by x. 



