﻿Summation 
  of 
  a 
  Type 
  of 
  Infinite 
  Series. 
  

   tx 
  + 
  /3 
  for 
  k 
  08=0, 
  1, 
  2, 
  . 
  . 
  ., 
  *-l), 
  and 
  have 
  

  

  899 
  

  

  K 
  = 
  S 
  'a+0 
  

  

  = 
  ^p^^y+^^+fj^^+o; 
  (3) 
  

  

  ad 
  

  

  -=— 
  — 
  - 
  f 
  dx 
  

   1-M* 
  

  

  = 
  f 
  S\- 
  l^+r- 
  1 
  **+' 
  ^ 
  + 
  ( 
  - 
  1)* 
  f^p 
  cfe 
  4- 
  C/' 
  . 
  . 
  (4) 
  

  

  Now 
  

  

  a# 
  l 
  r 
  i 
  

  

  1 
  — 
  x 
  c 
  ~ 
  "~ 
  t 
  \_x~l 
  

  

  # 
  + 
  1 
  

  

  2\tt\ 
  2\(£+l>r 
  

  

  2 
  cos 
  — 
  — 
  ) 
  cos 
  - 
  

  

  t-2 
  t-1 
  /_ 
  _ 
  2\7T\ 
  

  

  -^-° 
  r 
  -2- 
  (2x— 
  2 
  cos 
  -7-)- 
  

   , 
  ^ 
  \ 
  t 
  / 
  c 
  

  

  fc=i 
  # 
  2 
  — 
  2^ 
  cos 
  f- 
  1 
  

  

  *-2 
  *_i 
  . 
  2\tt 
  2a(^ 
  + 
  1)tt 
  _ 
  2A/3tt 
  

   -2T° 
  v 
  ^r 
  2 
  cos— 
  7- 
  cos 
  ; 
  2 
  COS 
  

  

  a.=i 
  a?- 
  4 
  — 
  2a? 
  cos 
  hi 
  J 
  

  

  By 
  means 
  of 
  (5) 
  we 
  obtain 
  from 
  (3) 
  

  

  a-l 
  #fy+/3+l 
  

  

  - 
  I 
  [log 
  (l-.r)+ 
  |(-1)3 
  [l 
  + 
  (-l)<] 
  log 
  (1 
  + 
  *) 
  

  

  *-2 
  f-1 
  

  

  + 
  2 
  cos 
  ~ 
  — 
  ^— 
  log 
  ( 
  x 
  2 
  — 
  2x 
  cos 
  + 
  1 
  ) 
  

  

  \=i 
  t 
  \ 
  t 
  J 
  

  

  t-2 
  t~\ 
  o 
  

  

  or 
  — 
  — 
  & 
  COS 
  

  

  .^oo. 
  2^+l> 
  _ 
  2 
  C0S 
  2_M!T 
  , 
  .»» 
  

  

  <£ 
  sin 
  

  

  + 
  2 
  9 
  > 
  -tan- 
  1 
  

  

  sm-— 
  ±— 
  #cos 
  

  

  1 
  t 
  

  

  3P2 
  

  

  an- 
  t 
  1 
  

  

  -, 
  2\tJ 
  

   1 
  — 
  o?cos 
  

  

  (6) 
  

  

  