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Question
Find the sum of n terms of the series 1+2x+3x2+4x3+5x4+....
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Solution
The series is the expansion of
(1−x)−2∴Sn=1+2x+3x2+....+nxn−1−2xSn=−2x−4x2−6x3−......−2(n−1)xn−1−2nxnx2Sn=x2+2x3+....+(n−2)xn−1+(x−1)xn+nxn+1;Sn(1−x)2=1−(n+1)xn+nxn+1∴Sn=1−xn(1−x)2−nxn1−x
The series is the expansion of
(1−x)−2
∴Sn=1+2x+3x2+....+nxn−1
−2xSn=−2x−4x2−6x3−......−2(n−1)xn−1−2nxn
x2Sn=x2+2x3+....+(n−2)xn−1+(x−1)xn+nxn+1;
Sn(1−x)2=1−(n+1)xn+nxn+1
∴Sn=1−xn(1−x)2−nxn1−x
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