Both of these questions could be true or false. I think I got the first one down, but the second I am going to need help on.
1) If \(\displaystyle a_k\le b_k\) for all \(\displaystyle k\in \mathbb{N}\) and \(\displaystyle \sum_{k=1}^\infty b_k\) is abs. convergent, then \(\displaystyle \sum_{k=1}^\infty a_k\) converges.
Proof of 1): I think this is false. My counterexample is if we let \(\displaystyle a_k = -1\) and \(\displaystyle b_k = 0\). So \(\displaystyle \sum_{k=1}^\infty b_k\) is abs. convergent but since \(\displaystyle a_k = -1\), \(\displaystyle \sum_{k=1}^\infty a_k\) diverges. Q.E.D. See any flaws in this counterex?
2) If \(\displaystyle \sum_{k=1}^\infty a_k\) is abs. convergent and \(\displaystyle a_k \downarrow 0\) as \(\displaystyle k\to \infty\), then \(\displaystyle \limsup_{k\to \infty} |a_k|^{1/k}<1\).
Proof of 2): ??? I have no clue even if it's true or false. First of all I do not understand the meaning of the down arrow in \(\displaystyle a_k \downarrow 0\). Any kind of help will go a long way into my understanding. Thanks.
1) If \(\displaystyle a_k\le b_k\) for all \(\displaystyle k\in \mathbb{N}\) and \(\displaystyle \sum_{k=1}^\infty b_k\) is abs. convergent, then \(\displaystyle \sum_{k=1}^\infty a_k\) converges.
Proof of 1): I think this is false. My counterexample is if we let \(\displaystyle a_k = -1\) and \(\displaystyle b_k = 0\). So \(\displaystyle \sum_{k=1}^\infty b_k\) is abs. convergent but since \(\displaystyle a_k = -1\), \(\displaystyle \sum_{k=1}^\infty a_k\) diverges. Q.E.D. See any flaws in this counterex?
2) If \(\displaystyle \sum_{k=1}^\infty a_k\) is abs. convergent and \(\displaystyle a_k \downarrow 0\) as \(\displaystyle k\to \infty\), then \(\displaystyle \limsup_{k\to \infty} |a_k|^{1/k}<1\).
Proof of 2): ??? I have no clue even if it's true or false. First of all I do not understand the meaning of the down arrow in \(\displaystyle a_k \downarrow 0\). Any kind of help will go a long way into my understanding. Thanks.