142 Mr. Moon on the Evaluation 



The Professor begins, "The general expression for the 

 sum of the infinite series 



1— x + x* — x 3 + x 4 — &c. 



1S B " TT^ ~ T+r 



What may be meant by the recondite symbol oo ", I do not 

 profess to understand. But it appears to me that 



« = -!_ (-*) 00 - 



1+x 1+x ' 

 and I am induced to think that Prof. Young himself will come 

 to be of the same opinion when he again undertakes to exa- 

 mine the subject. 



It is unnecessary for me to reply to the argument of the 

 present paper, which appears to rest upon one of Prof. Young's 

 previous fallacies which I have elsewhere exposed. I shall 

 merely advert to the conclusion at which he ultimately arrives, 

 " that it is indisputably true that the extreme of the convergent 

 cases of the above series S, usually written in the form 

 1 — 1 + 1—1 + &c. 



is — , and that the extreme of the divergent cases, usually 



written in the same form, is really infinite, as stated in his 

 former paper." 



I am afraid that Prof. Young will be apt to mystify both 

 himself and his readers by talking about "the extreme of the 

 divergent cases" and "the extreme of the convergent cases." 

 If these " extremes " are usually written under the above form, 

 I can only say that such usage is " extremely" improper. But 

 let Prof. Young define the terms he uses. What is meant by 

 "the extreme of the convergent series" for example? Is it 

 the value of the series 



l-(l-tf) + (l-:r) 2 +(l-.r) 3 -&c («.) 



when x — ? If so, I beg to assure him that the extreme 



value of the convergent series is not — , but 1 or indiffer- 



25 



ently. The fact is, it is absurd to talk of "extreme values" 

 in these cases. If x be made ever so small, there will al- 

 ways be some smaller value which might be assigned to it; 

 so that it is impossible to assign an extreme value to x so long 

 as it is an existing magnitude, and the moment it ceases to be 

 such the series ceases to be equal (or rather to approximate) 



to — . Twist and turn it as he may, Prof. Young will never 



be able to prove the series 



