Yesterday I was explaining the mechanics of numeric series to a friend of mine. When I was talking about the divergence of harmonic series $\sum_{n=1}^\infty\frac{1}{n}$ and the convergence of series $\sum_{n=1}^\infty\frac{1}{n^\alpha}$, for all $\alpha>1$ (Dirichlet's Theorem), she asked me if the harmonic series is the smallest divergent series that exists. Of course I said NO! For instance, the series $\sum_{n=2}^\infty\frac{1}{n}$, $\sum_{n=1}^\infty\frac{1}{3n}$ or $\sum_{n=1}^\infty\left(\frac{1}{n}-4\right)$ are all divergent and smaller than $\sum_{n=1}^\infty\frac{1}{n}$.
I know that my examples are correct, but she said (with some truth on her point of view), that I was cheeting and wasn't giving her a true reflected example of what she was asking.
Well, the truth is I can't imagine a proper example to give her!
Can anyone give me one? I'm (she is) looking for a divergent non rational nor irrational series $\sum_{n=1}^\infty u_n$
a) of non negative terms (at least from some $n_0$);
b) smaller than harmonic series $\sum_{n=1}^\infty\frac{1}{n}$.
c) $\lim_{n\to\infty}\sum_{p=1}^n\frac{1}{p}=+\infty$.
I think she will be pleased if I give her an example on the previous conditions.
Thanks.
Best Answer
What about $\displaystyle\sum_{n=2}^\infty\frac1{n\log n}$?