Let $a_n$ be a sequence of positive reals, such that the partial sums $S_n = \sum_{i = 1} ^ n a_i$ diverge to $\infty$. For given $\epsilon > 0$ do we have $$\sum_{n = 1} ^ \infty \frac{a_n}{S_n^{1 + \epsilon}} < \infty?$$
For $\epsilon \ge 1$ we can resolve this quickly by noting $$\frac{a_n}{S_n ^ 2} \le \frac 1 {S_{n – 1}} – \frac 1 {S_n}$$ so for sufficiently large $n$ we can bound $\frac{a_n}{S_n^{1 + \epsilon}}$ by $\frac 1 {S_{n – 1}} – \frac 1 {S_n}$ as well. I'm wondering if this is true for arbitrary $\epsilon > 0$. I know that the series in question diverges for $ \epsilon = 0$, so all that is missing is what happens in $(0, 1)$.
Best Answer
I proved something like this a while ago where I showed that if $0<a_{n-1}\le a_n$ and if $\epsilon>0$, then $$ \sum_{n=0}^\infty\frac{a_n-a_{n-1}}{a_n^{1+\epsilon}} $$ converges. I believe this is the same stiuation, where my $a_n$ is the $S_n$ in this problem. However, there is no requirement that $S_n$ (my $a_n$) diverges. Here is the proof I gave with my $a_n$ replaced by $S_n$.
By the Mean Value Theorem, for some $z_n$ between $S_{n-1}$ and $S_n$, we have $$ \frac{1}{S_{n-1}^\epsilon}-\frac{1}{S_n^\epsilon}=\epsilon\frac{S_n-S_{n-1}}{z_n^{1+\epsilon}} $$ Let us use this in the following telescoping series $$ \begin{align} \frac{1}{S_{k-1}^\epsilon}-\frac{1}{S_N^\epsilon} &=\sum_{n=k}^N\frac{1}{S_{n-1}^\epsilon}-\frac{1}{S_n^\epsilon}\\ &=\sum_{n=k}^N\;\epsilon\frac{S_n-S_{n-1}}{z_n^{1+\epsilon}}\\ &=\sum_{n=k}^N\;\epsilon\left(\frac{S_n}{z_n}\right)^{1+\epsilon}\;\frac{S_n-S_{n-1}}{S_n^{1+\epsilon}}\\ &\ge\epsilon\sum_{n=k}^N\frac{S_n-S_{n-1}}{S_n^{1+\epsilon}}\\ &=\epsilon\sum_{n=k}^N\frac{a_n}{S_n^{1+\epsilon}} \end{align} $$ This last inequality, along with the fact that $\frac{1}{S_n^\epsilon}$ is a non-increasing sequence bounded below by $0$, implies that the summation converges.