Question
The sum of the series 1 + 2x + 3
x2 + 4
x3 + ..........upto n
terms is
x2 + 4
x3 + ..........upto n
terms is
Answer: Option A
:
A
Let Sn be the sum of the given series to n terms, then
Sn = 1 + 2x + 3
x2 + 4
x3 + ....... + n
xn−1 ..........(i)
xSn = x + 2
x2 + 3
x2 + ...........+ n
xn ..........(ii)
Subtracting (ii) from (i), we get
(1-x)Sn = 1 + x +
x2 +
x3 + ......... to n terms -n
xn
= (
(1−xn)(1−x)) - nxn
⇒ Sn =
(1−xn)−nxn(1−x)(1−x)2 =
1−(n+1)xn+nxn+1(1−x)2
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:
A
Let Sn be the sum of the given series to n terms, then
Sn = 1 + 2x + 3
x2 + 4
x3 + ....... + n
xn−1 ..........(i)
xSn = x + 2
x2 + 3
x2 + ...........+ n
xn ..........(ii)
Subtracting (ii) from (i), we get
(1-x)Sn = 1 + x +
x2 +
x3 + ......... to n terms -n
xn
= (
(1−xn)(1−x)) - nxn
⇒ Sn =
(1−xn)−nxn(1−x)(1−x)2 =
1−(n+1)xn+nxn+1(1−x)2
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