26 Prof Young on the Summation of Series, 



which being equal to (D.), independently of particular values 

 of x, must be identical with it; hence the transformation (E.) 

 does not assist us in the summation of (D.). Slow series, how- 

 ever, of the form (D.) may, when b, c, d, &c. are a series of 

 divisors in arithmetical progression, be easily changed into 

 others, of quicker convergency, by a method which a single 

 example willfully illustrate. Suppose, for instance, the sum- 



X^ 1 ' X 



mation of x + — -f- — + — + &c. be required. Put 



/**3 ,v,5 y.7 



X X 3 X b X 1 , S,— X 



• • V- h u — - + &c. = —* . 



Subtracting 

 .r # 3 # 5 ^ 7 o 1 f o. S,— .rl 



Hence 



and consequently S x will be obtained by summing the more 

 rapid series S 3 . Again, 



A+A + 577+A + * --* 



x x* 47 x b jg S 3 -~ 



•'•3T5 + 5.7 + 1^9 + 9.11 +&c jnf--> 



Subtracting 



Hence S 3 = ^ > _ 1 = .p^y ( 4* S 5 - — J ; 



and thus S 3 , and therefore S 19 may be obtained by summing 

 the more converging series S 5 . And generally 



so that by summing a few terms of S„ we may, by this for- 

 mula, obtain near approximate values to S„_ 2 , S„_4, &c. in 

 succession, and finally, to Sj, each step being deduced from 

 the preceding by simple arithmetical operations. 



