Prove that the following sequence converges, and find the limit.?

an = [1/(n^3)] ∑ [k^2 cos(kπ/n)] , where the summation ∑ goes from k=1 to n.

I tried to show that the sequence was monotonically decreasing. I then wanted to use this along with showing a lower bound existed to prove that it was convergent, but I do not know how to go about this. Any help is appreciated.


Finding the limit is actually the easy part. If you write a_n as


Σ (k/n)2 cos(kπ/n) (1/n),


you can recognize this as a Riemann sum with an equally spaced partition of the interval [0,1]. Note that for

Δx = 1/n, x_k = k/n, and f(x) = x2 cos(πx),

in the limit as n->∞, the sum becomes


∫ x2 cos(πx) dx = -2/π2.


I'll have to try to work on the convergence argument---if one separate from actually finding the limit is necessary.

I hope this at least helps!