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Find the sum of series 4 - 9x + 16 x^(2)...

Find the sum of series `4 - 9x + 16 x^(2) - 25 x^(3) + 36 x^(4) - 49 x^(5) + ...oo`

Text Solution

Verified by Experts

Let `S = 4 - 9x + 16 x^(2) = 25 x^(3) + 36 x^(4) - 49^(5) + ....oo`
`-Sx = - 4x + 9x^(2) - 16 x^(3) + 25 x^(4) - 36 x^(5) + ...oo`
On substraction, we get
`S (1 + x) = 4 - 5x + 7x^(2) - 9x^(3) + 11 x^(4) - 13 x^(5) + ...oo`
`-S (1 + x)x = - 4 x + 5x^(2) - 7x^(3) + 9x^(4) - 11 x^(5) + ...oo`
On substraction, we get
`S (1 + x)^(2) = 4 - x + 2x^(2) - 2x^(3) + 2x^(4) - 2x^(5) + ... oo`
`- 4 - x + 2x^(2) (1 - x + x^(2) - ...oo) = 4 - x + (2x^(2))/(1 + x) = (4 + 3x + x^(2))/(1 + x)`
`S = (4 + 3x + x^(2))/((1 + x)^(3))`
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