30 Mr. W. Shanks on the [Nov. 23, 



that the sum of the series to 5000 terms has been found, and that the 

 series is required to be summed to 10,000 terms. 



To obtain the sum of the reciprocals of the even numbers from 5000 to 

 10,000, we have:— 



(-J L _J L 



l n 5 o i • 250 a ~ 

 ( FaTT TU 7T 3" ~f~ 



(dr + 



(sh + 



(xir + 



(tV+ 



These twelve quotients, when added together, give the valne of the reci- 

 procals of the even numbers between 5000 and 10,000, including the 

 latter number. 



To obtain the sum of the reciprocals of the odd composite numbers 

 between 5000 and 10,000, we have, using prime divisors, 



( 1 66 7 TdW * * ■ ' $~33l})~3' 



( 1 (A) 1 "t" 1 ti\) 3 1 00 7 * • ' • i99i))~ rD ' 



Here it must be observed that all odd numbers which are multiples of 

 previous prime divisors must be excluded : e. g. youT mus ^ he excluded 

 from division by 5, because 1005 is a multiple of 3. 



(l4»+ tAt)-7. ( T h+ *tt)**h 



(ih+ *§r-)*tt' (rhr+ Ar)-43. 



(*+ (t*t+ 



(t*i+ t4t)-17. (^ + fkH53. 



ibHUf (A + ii7)-59- 



7fcH23. (J ? + t*I>+«'- 



(t^+ *kH29. + tb>+«& 



(ih+ (7^ + ykH7i. 



(t£?+ ^)^37. 



Here it must be noted that all numbers below prime divisors must 

 always be excluded. 



Mr+ tW+73. (&+ ffc>+»< 



(tV+ Wr+ T kH97. 



(*+ Tk)-83. 



These twenty-four quotients, when added together, give the sum of the 

 reciprocals of the odd composite numbers between 5000 and 10,000. To 

 this sum add the sum of the prime reciprocals between 5000 and 10,000 ; 

 the result is the value of the reciprocals of all the odd numbers between 

 5000 and 10,000. It need scarcely be stated that the sum of these two 



•••i&*)-s-*. <*+ »V)+*. 



• ' ' T2V9) 2 ' 3 • (xi + T9 ) 29# 



•••«k)-2'. (1+ i)-2'». 



1 ~\ _l_ 9 5 L.i.9 11 



*"* 311/ * * 3 • * * 



. .. 1 4^)-r2 6 . l-f-2 13 . 



