Evaluate lim(n->inf) and prove your result.
Your summand is:
Now, of all of these the 1/n sum is going to be the most important because the sum of the other two as n approaches infinity converges, whereas the 1/n series does not. So your series does not converge.
There are innumerable ways to prove that the 1/n series does not converge and that the other two do so I leave this part of the proof to you.
-Dan
Evaluate:
Lim(n -> infty) [Sum(k=1..n) (n+2k)^2 / n^3]
First look at:
S(n)= Sum(k=1..n) (n+2k)^2 / n^3 = Sum(k=1..n) (n^2+4kn+4k^2) / n^3
.....=[Sum(k=1..n)1/n] + (4/n^2)[Sum(k=1..n)k] + (4/n^3)[Sum(k=1..n)(k^2)]
.....=1 + (4/n^2) n(n+1)/2 + (4/n^3) n(n+1)(2n+1)/6
.....= 1 + (2+2/n) + (4/3 +2/n +(2/3)/n^2)
Hence:
Lim(n -> infty) [Sum(k=1..n) (n+2k)^2 / n^3]=Lim(n -> infty) S(n) = 13/3
(But I would recomend that you check my working!)
RonL