Let $a_n > 0$ and for all $n$ let $\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $ Prove or give a counterexample to the statement $\sum\limits_{j=1}^{\infty} a_j < \infty$
Not sure where to start, a push in the right direction would be great. Thanks!
Let $a_n > 0$ and for all $n$ let $\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $ Prove or give a counterexample to the statement $\sum\limits_{j=1}^{\infty} a_j < \infty$
Not sure where to start, a push in the right direction would be great. Thanks!
Think about what bounds you can put on these:
$b_k = \sum\limits_{j=2^k}^{2^{k+1}-1} a_j$
and note that $\sum\limits_1^\infty a_n = \sum\limits_0^\infty b_k$.
Consider sum of sums $\sum_{i=1}^\infty \sum_{k=i}^{2k} a_j$.