13th Putnam 1953/B1 solution

Problem B1

Does ∑₁^∞ 1/n^{1 + 1/n} converge?

Solution

Answer: no.

x < e^x for all x ≥ 0, so x^1/x < e < 3. Hence 1/n^{1 + 1/n} > 1/3n. But ∑ 1/n diverges.