What is the limit of 1/n^2 + 2/n^2 + + n-1/n^2 as n-> infinity

[sara_87]

Homework Statement

find:

lim(n\rightarrow\infty (1/n^2 + 2/n^2 + 3/n^2 + ... + n-1/n^2 )

Homework Equations

3. The Attempt at a Solution [/b

I could guess that the limit is zero but i don't know howto prove it

 

[HallsofIvy]

Homework Statement

find:

lim(n\rightarrow\infty (1/n^2 + 2/n^2 + 3/n^2 + ... + n-1/n^2 )

Homework Equations

3. The Attempt at a Solution [/b

I could guess that the limit is zero but i don't know howto prove it

No, the limit is not 0.

That sum is the same as (1/n²)(1+ 2+ 3+ ...+ (n-1)). Can you write that last sum in closed form?

 

[sara_87]

Oh right,
so, this is the same as:

\frac{1}{n^2}\sumk (from k=1 to n-1)

and now i use the formula for geometric series?

 

[Mark44]

Dick asked whether you knew a formula for 1 + 2 + 3 + ... + (n - 1). This is not a geometric series.

 

[sara_87]

What is meant by 'closed form' ? i don't know, i thought dick meant that i should put it in a summation.
i meant arithmetic...sorry, surely this is not geometric.

 

[sara_87]

I have done it and i just want to make sure of my answer:
i used the summation formula for aritmetic series :1/2 n(n+1)
and got that the limit is 1/2
is that right?

 

[Dick]

I have done it and i just want to make sure of my answer:
i used the summation formula for aritmetic series :1/2 n(n+1)
and got that the limit is 1/2
is that right?

Sure it's right. You could also look at the problem as being a Riemann sum for the integral of f(x)=x from x=0 to x=1. Surely, 1/2. BTW, I didn't say anything before in this thread. That was Halls. Let's give credit where credit is due.

 

[Mark44]

BTW, I didn't say anything before in this thread. That was Halls. Let's give credit where credit is due.

Sorry for giving credit where not due. I think I saw your name in another thread that sara started, and mistakenly cited you and Dirk.

 

[Dick]

Sorry for giving credit where not due. I think I saw your name in another thread that sara started, and mistakenly cited you and Dirk.

No problem. If Hall's suggestion had been wrong, I would have been OUTRAGED to have it attributed to me. But it wasn't. :)