= Expectation of an absolutely summable random series
{title2=$E[\sum_iX_i]=\sum_iE[X_i]\quad\text{if }\sum_iE|X_i|<\infty$}
If <random variables> $X_i$ satisfy $\sum_iE|X_i|<\infty$, then $\sum_iX_i$ converges absolutely <almost surely>, is integrable, and its <expected value> equals $\sum_iE[X_i]$. Indeed, the <monotone convergence theorem> gives $E[\sum_i|X_i|]=\sum_iE|X_i|<\infty$, so that nonnegative sum is finite <almost surely>. It dominates every finite partial sum in absolute value; the <dominated convergence theorem> then justifies passing the <limit> through the <expected value>. No <independence> is required.
For example, if $|X_i|\le Mq^i$ for a fixed $M$ and $0<q<1$, the <geometric series> bound proves the hypothesis directly. This is useful when repeated observations are multiplied by geometrically shrinking distances or weights.
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