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`

Answer

`0017`

Explanation

Consider
`sum_(k=1)^(n)(2k(k^(2) + 4k+3)-1)/(k^(2) + 4k+3) = sum(2k) -(1)/((k+1)(k+3))`
` =2sum k-(1)/(2) sum((1)/(k+1)-(1)/(k+3))`
`=(2(n)(n-1))/(2)(1)/(2)[(1)/(2) + (1)/(3)] = n^(2) +n-underset(n to oo)ubrace((1)/(2)[(1)/(2) + (1)/(3)])`
Hence , Sum ` =(n^(2) + n) -(n^(2) + n-(5)/(12)) =(5)/(12)`
`implies p=5, q=12 " "implies (p+q) = 17`