316 Mr. H. A. Webb. On the Convergence of [Nov. 10, 



By means of this theorem approximate expressions for large values 

 of one or more of the arguments can be found for many functions that 

 occur in analysis and satisfy linear differential equations of the second 

 order. For instance, the approximate solution of Lamp's equation 



for large values of n is 



sn~k [Ci (dnz + kcnz) tt+ l + C 2 (d 



where Ci and C 2 are arbitrary constants. 



We can often determine the condition of convergence of an infinite 

 series of functions by replacing the nth term by its approximate value 

 for large values of n. For instance, the series of Lame's functions* 



l<VE(8ina;).F*(sin/), 



n=l 



where the c's are arbitrary constants, subject to the condition that the 

 series 



has unit radius of convergence, converges if 



) dn (ft, k) ._dn (r^V) 

 " 



_ 



en (ft*) en (ft,*)" 



here 



The limitation of the method is noteworthy; we cannot find the 

 condition of convergence of the expansion of a given function in an 

 infinite series of given functions without knowing 



(i) That the expansion formally exists ; and 



(ii) The approximate, if not the accurate, value of the ?ith term in 

 the expansion for large values of n. 



In the case of an expansion in hypergeometric functions, the limita- 

 tion may be removed with the help of two theorems : 



(i) If < (z) be a solution of the linear differential equation of the nth 

 order, 



the coefficient of -^ being a polynomial in z of order r ; then 



* The notation is that of Byerly ('Fourier's Series and Spherical Harmonics ' 

 p. 266 (1895)), B=j> (! + *). 



