Determine whether the sequence converges or diverges and find the limit

[Sean1218]

Homework Statement

Determine whether the sequence a_n = 1¹/n² = 2¹/n² + ... + n¹/n² converges or diverges. If it converges, find the limit.

2. The attempt at a solution

I have no idea what to do with this problem. I don't see why I can't simplify n/n^2 to 1/n. It was suggested to me to factor out 1/n and introduce the variable i (from Riemann Sums), but I don't see how that helps (and I don't see how I would just introduce i anyway).

 

[Dick]

Factor out the 1/n^2. Can you write an expression for the sum (1+2+...+n)?

 

[Sean1218]

I'm not really sure what you mean.

 

[Dick]

I'm not really sure what you mean.

I meant pretty much what I said. an=(1/n^2)*(1+2+...+n), right? Perhaps you know a formula for (1+2+...+n) in terms of n? If not, and you know how to integrate you can also do it as a Riemann sum. Can you write down a Riemann sum for the function f(x)=x between 0 and 1?