Given that $a_n, b_n > 0 $ and that $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ both converge, does the property hold? Is there a particular case when it is true like maybe when $\sum_{n=1}^\infty (a_n-b_n)$ is absolutely convergent?

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    $\begingroup$ The equality is true if $\sum a_n$ and $\sum b_n$ are (not necessarily absolutely) convergent. This is a simple consequence of linearity of sum. See this answer for a proof. $\endgroup$ – Sangchul Lee Jan 22 at 1:50

If $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ both converge, then yes the equality must hold. This is a consequence of properties of limits. Namely, that if $\lim_{n \to \infty} s_n$ and $\lim_{n\to \infty} t_n$ both exist, then $\lim_{n\to \infty} (s_n-t_n)$ exists and equals the difference of the limits. So,

\begin{align*} \sum_{n=1}^\infty (a_n-b_n)&=\lim_{N\to \infty} \sum_{n=1}^N (a_n-b_n)\\ &= \lim_{N \to \infty} \sum_{n=1}^N a_n - \lim_{N\to\infty} \sum_{n=1}^N b_n\\ &=\sum_{n=1}^\infty a_n - \sum_{n=1}^\infty b_n. \end{align*}


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