S1=1−1+1−1+1−1+1−...=0[if ,n=even]
S1=0+1−1+1−1+1−1+...=1[if n=odd]
now add (i) and (ii)
∴2S1=1
∴S1=21
S2=1−2+3−4+5−6+...
S2=0+1−2+3−4+5−...
nowadd(iii)and(iv)2S2=1−1+1−1+1−...∴2S2=S1=21∴S2=41S3=1+2+3+4+5+...S2=1−2+3−4+5−...∴S3−S2=0+4−0+8−0+16−0+20−...∴S3−41=4(1+2+3+4+5+...)∴4S3−S3=−41∴S3=−121∴1+2+3+4+5+...=−121
