2— NUMERICAL SUMMATION OF THE RECIPROCALS 

 OF THE NATURAL NUMBERS. 



By R. T. a. Innes, F.R.A.S. 



[Abstract.] 



To obtain the sums of the fractions j to ^, ^ to g\,, and so on, 

 the summation formula 



2 - = C + loge p H • — .. + i - ^^ — 6 + &c... (a) 



p ^"^ ^ ' 2p I2p" I20p^ 252P'' ^ ' 



could be used. It will, however, be better to use a modification of 

 a formula given by Schlomilch (Compendium II., 1895, p. 234). 

 It is as follows : — 



J 10 -I Bt-io^'-i B3 10^ - I 

 ^irr-£ = loge 10 + aT^TT+T +y i^a^:^) "4710^'"+''''' "' 



wherein Bi, Bj, etc., are Bernouilli's Numbers. 



For numerical application this formula may be written (n = o 

 being excluded) : — 



2 ; 2 ;; — t = Og, 10 H x 0.4S 



10"+^ -I ^10" -I ^' 10" ^ 



+ ^uX 0-0825 

 -^^x 0.0083325 



+ ^nX 0.00396825... 



I 



-;^x 0.004132231... 



+ &c. 



Thus, it is seen that as n increases, the sums of the parts tend 

 towards the limit loge 10 = 2.302585093. 



The sums of the earlier parts of this infinite series are : — 

 I to ^ = 2.8289683 



= 2.3484093 

 = 2.3070933 

 = 2.3030352 

 = 2.3026301 

 = 2.3025896 



Adding these parts together and increasing the total by one-millionth, 

 we have for the sum of the reciprocals of the natural numbers up 

 to one million, or 



I + i + i to THsU = 14-3927268 



The direct use of formula (a) yields 14.3927267, an agreement 

 within the chosen limit of accuracy. 



