# When is $\sum_{n=1}^\infty (a_n-b_n) = \sum_{n=1}^\infty a_n - \sum_{n=1}^\infty b_n$ true?

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?

• 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. Commented Jan 22, 2019 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*}