Let Sk be the sum of an infinite GP series whose first term is k and the common ratio is kk+1(k>0). Then, the value of ∑k=1∞(-1)kSk is equal to
loge4
loge2-1
1-loge2
1-loge4
Determine the value of ∑k=1∞(-1)kSk
We know that sum of an infinite GP series is given by: S=a1-r
Sk=k1-kk+1∵commonratio=kk+1,Firstterm=k,given=k(k+1)k+1-k=k(k+1)
Now,
∑k=1∞(-1)kSk=-11.2+12.3-13.4..........∞=-(1-12)+(12-13)+(-1)(13-14)......∞=-1+12+12-13-13.........∞=-1+212-13+14.........∞=-1-2-12+13-14.........∞=-1-21-12+13-14.........∞+2Since,loge2=1-12+13-14.........∞∴1-2loge2or1-loge4
Thus, the correct option is D.