Question

Determine whether the sequence an=n312+n322++n3n2 converges or diverges. If it converges, find the limit.

Answer 1

Perhaps it suffices to say that

`\displaystyle\frac1{1^2}+\cdots+\frac1{n^2}=\sum_(k=1)^n\frac1{k^2}`

converges to
`\frac{\pi^2}6` as
`n\to\infty`, while
`n^3\to\infty`. Then

`\displaystyle\lim_(n\to\infty)a_n=\infty\cdot\frac{\pi^2}6=\infty`

so
`a_n` diverges.