Here is another self-study exercise that I am struggling mightily with:

$X_n=\sum\limits_{k=1}^n\frac{(-1)^k}{k^2}x_k$ where $\omega=(x_1,x_2,...)$ is a series of Bernoulli (1/2) trials.

I am told that $X_n\to X$ a.s for some $X$, and am to show whether $E(X_n)\to E(X)$ as $n\to\infty$

I do not need to explicitly calculate the expectation, but just show its convergence, if applicable.

As I get more and more familiar with dominated convergence, monotone convergence, Fatou, etc. I may not need as much explicit help, but in this exercise if you could help me identify which of the convergence theorems is necessary (and hints at the justification for such), it would be of great help.

  • For the first part, use the fact that $|(-1)^kx_k|=1$ to show the convergence of the series.
  • For the second one, use the fact that the convergence of partial sums is also in $L^1$.

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