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> Suppose I have an i.i.d sequence of random variables $X\_1,X\_2 \ldots $ such that $\mathbb{P}(X\_i =+1)=\mathbb{P}(X\_i = -1)=0.5$. I need to prove that the random series $\sum\_{k \geq 1} \frac{X\_k}{k}$ converges a.s. and in $L^1$.
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I should use the theorem that states that if I have $(M\_n)\_n$ martingale w.r.t $(F\_n)\_n $ filtration and let $F\_{\infty}=\sigma (F\_n:n \in \mathbb{N})$, then $(M\_n)\_n$ is uniformly integrable iff there exists a random variable $F\_{\infty}$-measurable $M$, such that $M\_n \rightarrow M$ a.s. and in $L^1$.

So I should just check uniform integrability. From the text of the exercise, I think that the "obvious" martingale would be $M\_n=\sum\_{k=1}^{n} \frac{X\_k}{k}$. 

Now I get stuck, because I tried to check uniform integrability by checking that $\sup\_n E[|M\_n|] < +\infty$, but if I bound those sums I have:

$E[|M\_n|] \leq \sum\_{k=1}^{n} \frac{1}{k}E[|X\_k|]= \sum\_{k=1}^{n} \frac{1}{k}$ and this diverges as $n$ goes to $+\infty$. 

What am I missing in uniform integrability?