INFINITE SERIES 113 



TRANSFORMATION OF INFINITE SERIES 



Series which converge slowly may often be transformed to more rapidly 

 converging series by the following methods. 



6.20 Euler's transformation formula: 



S = do + a\x + a%x 2 + 



I X 



where: A#o = a\ a Q , 



- A 2 a = a 3 - 3 2 + 3^1 - a Q , 



The second series may converge more rapidly than the first. 

 Example i. 



*' 



-3'S- .(2*+ I) 



. jfe=0 



Example 2. 



l k? 



k=i 



6.21 MarkofT's transformation formula. (Differenzenrechmmg, p. 180.) 



' - 



~ i 



k=o k=o k=o k=o 



