Convergence of sum of random Poisson variables with divergent parameter I'm studying almost-surely convergence and convergence in probability of $S_{n}=X_{1}+\cdots+X_{n},$ where $X_{n}$ is distributed $\mathrm{ Poisson}\left(1/n\right)$ with $n\in\mathbb{N}$ and the sequence of $X_{n}$ are independent.

I'm stuck with this because I cannot applied Chebyshev inequality because of divergence of $\displaystyle\sum_{i=1}^{\infty}\frac{1}{n}$, neither Borel-Cantelli Lemma because I don't find a useful bound to use it. 
I tried use the convergence of $\displaystyle\sum_{n=1}^{\infty}E\left(X_n\right)<\infty$ implies $\displaystyle\sum_{n=1}^{\infty}X_n$ convergence a.s., but I don't get it.
Any kind of help is very thanked.
 A: Let $a_n:=\Pr\left(S_{2n}-S_n\geqslant 1\right)$. Using independence, we can see that $S_{2n}-S_n$ is Poisson distributed with parameter $\sum_{i=n+1}^{2n}1/i$. As a consequence, we have 
$$a_n\geqslant 1-\exp\left(-\sum_{i=n+1}^{2n}1/i\right) \geqslant 1-\exp\left(-\frac 12\right)\gt 0.$$
Therefore, the sequence $\left(S_n\right)_{n\geqslant 1}$ cannot be convergent in probability. 
A: The (finite) sum of independent Poisson distributed random variables is Poisson distributed with a parameter which is the sum of the individual parameters.
So $S_n$ is Poisson distributed with harmonic number parameter (and mean and variance) $H(n) = \displaystyle\sum_{m=1}^{n}\frac{1}{m} \approx \log_e{n}+\gamma+\frac{1}{2n}- \cdots$.
The only real convergence here is that $\dfrac{S_n}{H(n)} \to 1$ in probability and almost surely in a law of large numbers sort of sense and  $\dfrac{S_n}{\sqrt{H(n)}} -\sqrt{H(n)} \xrightarrow{d} N(0,1)$ in a central limit theorem sort of sense.  
The infinite series is almost surely greater than any given number.
