Determine whether the sequence an=12n3+22n3+⋯+n2n3 converges or diverges. if it converges, find the limit. note, ∑i=1ki2=k(k+1)(2k+1)6

I guess the sequence is

which we can write as

converges if it is bounded and monotonic. Consider the function,

which has derivative

for all , so is monotonically decreasing on , and as we have

So we know that is monotonically decreasing and bounded below by .

To find the limit, we can also write

As , the rational terms vanish and we're left with

2nd one in the first row is correct

where are the choices?