Putnam 1982/A2 solution

43rd Putnam 1982

Problem A2

Let B_n(x) = 1^x + 2^x + ... + n^x and let f(n) = B_n(log_n2) / (n log₂n)². Does f(2) + f(3) + f(4) + ... converge?

Solution

Answer: yes.

This looks much worse than it is. A crude estimate for B_n(log_n2) is sufficient. We know that ∑ 1/n only just fails to converge. So it is likely that ∑ 1/n 1/log²n will converge. In that case we need only prove that B_n(log_n2) ≤ O(n). But it has n terms, the biggest of which is 2. So certainly B_n(log_n2) < 2n.

We can easily check that ∑ 1/n 1/log²n converges by the integral test (the indefinite integral is -1/log n).

43rd Putnam 1982