Limit of ∑ k/n^2 from k=1 to n as n Approaches Infinity

[ƒ(x)]

If n is a positive integer, then the limit of \sum k/n^2 from k=1 to n as n approaches infinity = ?

A) pi
B) 1
C) 1/2
D) 1/pi
E) 0

How do I do this?

 

[dextercioby]

Can you compute the sum ?

 

[payumooli]

c) 1/2

since 1/n^2 is constant remove it outside
sum of n natural numbers formula can be used for the summation
then you have lim tending to infinity
n(n+1)/(2n^2)
use L 'opital

should be right

 

[LCKurtz]

c) 1/2

since 1/n^2 is constant remove it outside
sum of n natural numbers formula can be used for the summation
then you have lim tending to infinity
n(n+1)/(2n^2)
use L 'opital

should be right

Except that, technically, L'Hospital's rule does not apply since n is given to have integer values. Just divide the numerator and denominator by n² and use basic properties of limits.

 

[HallsofIvy]

If a sequence of functions, f(x), has limit L as x goes to a, then any sequence, {f(x_n)}, with {x_n} converging to a, must also converge to L. In particular, if f(x) goes to L as x goes to infinity, the sequence {f(n)} also converges to L. As long as a function of n, for n a positive integer, can be as a function of x, a real variable, (for example, does NOT involve factorials), L'Hopital's rule can be applied.

 

[LCKurtz]

Yes, of course. That's why I said "technically". Still, I think it is best if students learn to use the more elementary methods when they are appropriate.

 

[payumooli]

Yes, of course. That's why I said "technically". Still, I think it is best if students learn to use the more elementary methods when they are appropriate.

personally i wouldn't prefer L opital either

 

[ƒ(x)]

Thanks, the answer is C. But, is there another way to do this?

 

[LCKurtz]

Thanks, the answer is C. But, is there another way to do this?

You have been given two methods. Did you understand both? An experienced person doing a multiple choice question where you didn't have to show work or give a reason would have would just have looked at the ratio of the n² terms to get the answer.