Prove the following: Product of Roots

Question

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges

well I don't really know if it does but my gut tells me it does:

I can take the log of this product

to create:

$\ln(1) + \frac{\ln(2)}{2} + \frac{\ln(3)}{3} + \frac{\ln(4)}{4} \ldots$

which I believe is asymptotic to the integral from $\int_1^{\infty} \ln(x)/x \ dx$ which is equivalent to:

$$\lim_{c \rightarrow \infty} \ 1/2 \ln(c)^2 - 0 $$

= infinity?

Is the asymptotic assumption correct? If not then how to prove the convergence or divergence of this series?

Answer 1

Your reasoning works just fine. An easier way to see the the series diverges is to see that it's larger than the harmonic series.

Answer 2

There are various ways to do this, and the simplest is probably this $\sum_n \frac{\ln n}{n} \geq \sum_n \frac{1}{n} = \infty$, as the later is the harmonic series.