Let $ \sum_{n=1}^\infty a_n $ be a conditionally convergent series. Then the series $\sum_{n=1}^\infty b_n$ of positive terms of $\{a_n\}_{n=1}^\infty$ and $\sum_{n=1}^\infty c_n$ of negative terms are divergent.
My proof: if $\sum_{n=1}^\infty a_n$ is conditionally convergent, then $\sum_{n=1}^\infty |a_n| = \infty$. Let's suppose that $\sum_{n=1}^\infty b_n$ and $\sum_{n=1}^\infty c_n$ are convergent, so since they are composed of positive and negative terms respectively, then $\sum_{n=1}^\infty |b_n| < \infty$ and $\sum_{n=1}^\infty |c_n| <\infty$ and we have that $$ \sum_{n=1}^\infty |a_n| = \sum_{n=1}^\infty |b_n| + \sum_{n=1}^\infty |c_n| < \infty $$ which is absurd since we supposed that the series was conditionally convergent.
Edit
Can I define $b_n := \left\{\begin{array}{ll} a_n &\text{if } a_n > 0 \\ 0 &\text{otherwise} \end{array} \right.$ and $c_n$ analogously for negative terms and say that $\sum a_n = \sum b_n + \sum c_n $ and, since $\sum a_n$ converges then both $\sum b_n, \sum c_n$ converge? Are the new series equal to the series of positive (and negative) terms?
Is this correct? Thanks in advance
Best Answer
Note that $$b_n=\dfrac{a_n+|a_n|}{2} \;\;(\geqslant 0), \\ c_n=\dfrac{a_n-|a_n|}{2} \;\;(\leqslant 0) $$ and $$c_n=a_n-b_n.$$ If $\sum a_n$ converges (conditionally) and $\sum b_n$ is convergent (absolutely) then $\sum {c_n}$ is convergent (absolutely). Because $a_n=b_n+c_n$ then $\sum a_n$ must be absolutely convergent, which contradicts to its conditional convergence.