problem about almost sure convergence

[yavanna]

Does $\displaystyle (X_{n})_{n_{\geq 1}}$, $\displaystyle X_{n}=\frac{n}{ln(n)}Y_{n}$, where $\displaystyle Y_{n} \sim Exp(n)$ converge a.s. to 0?

 

[chrisr]

As n goes to zero, for n rational, this converges on 0,
but surely you mean for n natural.
if n>1, the function diverges, as n increases,
if Y = e[sup:152he7pm]-n[/sup:152he7pm], the function converges on zero as n increases.