352 



Prof. Young on the Summation 



+ 1 



+ 



•3 



•2 



•142857 



•111111 



•090909 



'4-4012 = sum of first six terms. 



10256 

 7844 

 6191 

 5013 



2412 

 1653 

 1178 



759 

 475 



284 



-H -076923 



— -066667 

 + -058823 



— -052632 

 + -047619 



The numbers -076923, 10256, 2412, and 284 are the re- 

 spective values of 6, A, A^ A^ and A'^in (C), and to deduce 

 from these the value of S, it will be necessary merely to add 

 half the last number 284 to the preceding, half the sum to 

 the next, and so on, to the last, adding half the final sum to 

 the number '744012 previously found. The remainder of 

 the operation, therefore, is as follows : 



•076923 10256 



•082767 11688 



-041384 5844 

 -744012 



2412 

 2864 

 1432 



759 

 901 

 452 



284 

 142 



•785396 = inferior limit. 

 142 



4 = 



•785400 = superior limit. 

 -785398 = -^ very nearly. 



This value of -— we know from the small interval between 

 4 



the limits cannot possibly differ from the truth by more than a 

 unit in the final decimal. It is, in fact, true even in the last 

 decimal. We have separated the two parts of the process in 

 this example for the purpose of clearer illustration; but they 

 may be combined as in the following. 



2. To find the value of the converging series 



3^ 3^.5" 3^ 5^ 1^ 



"T 7r3 Ti ^9 aq. ^9 "I ;T) ^j ,.j -^o — oCC, 



1 



2^ 



2^4" 2^4^6' 2^4^ 6^8^ 



-which occurs in the expression for the time of vibration of a 

 pendulum in a circular arc, 



