A RIGOROUS ELEMENTARY PROOF OF THE BINO- 

 MIAL THEOREM. 



BY FRANK H. LOUD. 



The text-books of Algebra in general use in the colleges of 

 this country employ, as a means of treating some of the subjects 

 presented, a few well-known infinite series, of which the most 

 necessary are the binomial formula, the logarithmic and the 

 exponential series. Taylor's or McLaurin's theorem, or both, 

 are often added, or sometimes made the basis of the proof given 

 for the others. Or the binomial formula may be used as the 

 fundamental theorem, and from it may be deduced by rigorous 

 proofs whatever series are required in the treatment of logar- 

 ithms or of higher equations, in so far as these subjects are 

 usually discussed in elementary works. A chapter on the .theory 

 of infinite series necessarily accompanies the foregoing, in which 

 the student is made acquainted with the idea of convergence, 

 and furnished with the more simple tests, by which the conver- 

 gence of series may in many cases be proved. 



But the discussion of infinite series in these text-books is 

 not only elementary (as a matter of course), but also to such a 

 degree incomplete, that in almost every case it fails to warrant 

 the use made in the text-book itself of the series employed; so 

 that it is not too much to say that all our text-books, except a 

 very few which are so difficult as to be seldom used, offer inac- 

 curate proofs of the fundamental theorems above mentioned, 

 thus offering to the student in lieu of a demonstration an argu- 

 ment that is in part fallacious.* 



*To the general prevalence of this defect in our college Algebras, the attention of the 

 writer was called by Prof. F. Cajori, to whose suggestions and criticisms, in aid of the 

 present attempt to supply that defect, a debt is due which it is a pleasure to acknowl- 

 edge. 



