0
$\begingroup$

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?

$\endgroup$
  • 5
    $\begingroup$ 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
2
$\begingroup$

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*}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.