92 G. FLEURI. 



There is a very important case when that condition of con- 

 vergence is fulfilled ; it is when the series of tensors 



T=^a 2 + b 2 + c 2 +^a 2 +6 a +c 2 + ^a 2 +b *+c * + ... 

 111 222 n n n 



is converging ; for, in that case, the terms of A, B and C are 

 separately smaller than those of the converging series T. 



Then the series 12 is said to be absolutely convergent* 



If now we consider a series ordered with regard to increasing 

 powers of a variable vector v 



12 = a + a x v + a 2 v 2 + ... +a n v n + ... 

 a oi «!, a 2 ...a n being constant scalar quantities, then t being the 

 tensor of v 



v = t (cos a + A sin a) 

 the series of tensors 



«o + a i t + a* t 2 + ...+a n t n +... 

 is converging when 



t < lim J^l. for n = oo 



a n +l 



Let lim an = R, then the condition of convergence is 

 t < R 

 or in geometrical language, the point determined by the vector 

 (variable) v drawn from an origin o must be interior to a sphere 

 of radius R which we may call sphere of convergence of the series. 



As an example let us consider the series 



l+JL+£ + ...+£+... 



1 2! m ^ 



which by natural extension of algebraic notation we will represent 

 by e v ; in that case 



*±- = n+ 1 so that lim -°^_ = oo 

 an+l an + l 



11=00 



therefore R = oo that is to say the series e v is absolutely converg- 

 ent for the whole space. 



Now consider the series e aA 

 where a is a scalar and A a unit vector we have 



* Of course a series of vectors may be convergent but not absolutely 

 convergent. 



