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Question
Let Sk be the sum of an infinite GP series whose first term is k and common ratio is kk+1(k>0). Then, the value of ∞∑k=1(−1)kSk is equal to
Let Sk be the sum of an infinite GP series whose first term is k and common ratio is kk+1(k>0). Then, the value of ∞∑k=1(−1)kSk is equal to
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Solution
The correct option is D 1−loge4
S1=1+12+122+∞=11−12=2
S2=2+2⋅23+2(23)2+......∞=21−23=6
S3=3+3(34)+3(34)2+∞=31−34=12
S4=4+4(45)+4(45)2+∞=41−45=20
∴∞∑k=1(−1)kSk=−1S1+1S2−1S3+1S4−∞
=−12+16−12+120−∞
=−11.2+12.3−13.4+14.5−∞$
=−(1−12)+(12−13)−(13−14)+(14−15)−.......∞
=−1+2(12−13+14−15+∞)
=−1−2(1−12−13+14−15+∞)+2
=−1−2loge2+2=1−loge4
S1=1+12+122+∞=11−12=2
S2=2+2⋅23+2(23)2+......∞=21−23=6
S3=3+3(34)+3(34)2+∞=31−34=12
S4=4+4(45)+4(45)2+∞=41−45=20
∴∞∑k=1(−1)kSk=−1S1+1S2−1S3+1S4−∞
=−12+16−12+120−∞
=−11.2+12.3−13.4+14.5−∞$
=−(1−12)+(12−13)−(13−14)+(14−15)−.......∞
=−1+2(12−13+14−15+∞)
=−1−2(1−12−13+14−15+∞)+2
=−1−2loge2+2=1−loge4
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