﻿Summation 
  of 
  a 
  Type 
  of 
  Infinite 
  Series. 
  901 
  

  

  The 
  following 
  example 
  will 
  illustrate 
  the 
  method 
  o£ 
  work. 
  

  

  To 
  find 
  oo 
  ..» 
  

  

  8 
  = 
  2-^—^ 
  

  

  n=0 
  n 
  (4n 
  + 
  m) 
  

  

  Then 
  

  

  [4 
  K=0 
  V 
  y 
  \/c/„= 
  4:7i 
  + 
  «+l 
  

   Let 
  r 
  = 
  ^ 
  4 
  , 
  then 
  

  

  S 
  =iil„<-^C)^ 
  S 
  «- 
  • 
  • 
  • 
  (1 
  > 
  

  

  wherein 
  

  

  oo 
  ^n+zc+l 
  f* 
  g, 
  K 
  

  

  n=0 
  4?i 
  + 
  «+l 
  Jl— 
  x* 
  

   We 
  have 
  

  

  S 
  = 
  ^ 
  tan 
  a? 
  + 
  - 
  log 
  j— 
  -, 
  

  

  Ol 
  = 
  j 
  log 
  

  

  4""°l-r' 
  

  

  1 
  , 
  1 
  + 
  ^ 
  1 
  , 
  _i 
  

  

  fe 
  9 
  = 
  - 
  log 
  - 
  — 
  - 
  tan 
  l 
  x, 
  

  

  4 
  6 
  1 
  — 
  x 
  2 
  

  

  5 
  3 
  = 
  jlog(l-^), 
  

  

  1 
  1 
  1+ 
  a? 
  

  

  5 
  4 
  = 
  ^tan" 
  1 
  ^ 
  + 
  jlogj^-"*. 
  

  

  By 
  means 
  of 
  these 
  results 
  (1) 
  becomes, 
  after 
  replacing 
  

   # 
  by 
  r 
  1/4 
  , 
  

  

  J^ 
  p— 
  6^/2 
  + 
  1 
  tan" 
  1 
  r 
  1 
  / 
  4 
  y 
  + 
  6r 
  1/2 
  +l 
  1 
  1+r 
  1 
  / 
  4 
  

  

  s 
  

  

  l 
  + 
  r 
  l/2 
  

  

  log-(l 
  + 
  r»fl)-i-i^Iog(l-^ 
  - 
  i] 
  . 
  (2) 
  

  

  If 
  s= 
  | 
  5 
  (-D^ 
  f 
  r>0? 
  

   nr=0 
  n 
  (4?i 
  + 
  m) 
  

  

  1 
  4 
  /AX 
  1 
  a, 
  r 
  4rc 
  + 
  /c+l 
  

  

  8 
  =ii? 
  <-H)^-nfr^r- 
  ^ 
  

  

  herein 
  x 
  = 
  r 
  1/4 
  . 
  

  

  