and 



or 



Summation of certain Types of Series. 79 



Now 73 c. 1 



S=J[ioga+^)~io g (i-^)] 5 



^ = |[(l + ^)loa(l + ^)+(l-^)log(l- A .)], 



Si= ^B (i+ ^ )io - (i+ ^-^ i+ ^ 2 



1-Kr 2 , 1 + ^ a? # 



= ^ 1 °gT^. + 2 1 °g( 1 -^-2- 

 Therefore 



Q l'+r, 1 + ri , 1_ M . 1 



S=^[ 1 -?loggJ + 21o g (l-,)-2]. 



If r is negative, that is, if we are to find 



o 1 1 1 2 _ 2 (-r)» 



"1.2.3 3.4.5 r+ 5.6.7 r "' n Z (2n-hl)(2n + 2)(2n+xy 



we must substitute in the results obtained above 



.i 1-4- if* i 1 + f 5 



if* for r*. and log- — , = 2itan _1 r 5 for logi r- 



& 1 — zr* & 1 — 7*2 



We then obtain 



S = - © [ 2 -^ta n -M + 21og(l + r)-2], 



= ^[l_log(l+r)-^tan-M]. 



III. To find 



»-i 



S = l+^r +I i4ir»+ 4 ' 5 - 6 r» + ...= ig^ft 



-n(ii+«) 



n=0 



(i.) First method : 



n-l 



n o (4+*) 10I (|i+3) j 10! x 



II (ii+/c) n (?i+tc) 



K=0 k=4 



