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If the value of the sum n^(2) + n - sum...

If the value of the sum `n^(2) + n - sum_(k = 1)^(n) (2k^(3)+ 8k^(2) + 6k - 1)/(k^(2) + 4k + 3)` as n tends to infinity can be expressed in the form `(p)/(q)` find the least value of (p + q) where p, q `in N`

Text Solution

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0017
Consider
`sum_(k=1)^(n) (2k(k^(2) + 4k + 3) -1)/(k^(2) + 4k + 3) = sum(2k) - (1)/((k + 1)(k + 3))`
`2sumk - (1)/(2) sum((1)/(k + 1) - (1)/(k +3))`
`= (2(n)(n -1))/(2) (1)/(2) [(1)/(2) + (1)/(3)] = n^(2) + n - (1)/(2) [(1)/(2) + (1)/(3)]_(n to oo)`
Hence , sum = `(n^(2) + n) - (n^(2) + n - (5)/(12)) = (5)/(12)`
`rArr P = 5 , q = 12 rArr (p +q) = 17`
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