ca.analysis and odes - Positivity of "harmonic" summation

You can get counter examples for various $alpha$ of the following form. Let

$$a_1=1, a_2=-2 + 2 epsilon, a_3=0, a_4=a, a_5 = -b.$$

Here $0<epsilon <1$ and we take $a$ is big enough so that $a + 2epsilon -1 ge 1$. Here is one such counterexample

$$a_1 = 1, a_2 = -frac{7}{4}, a_3=0, a_4=3, a_5= -frac{71}{16}.$$

The sequence of partial sums is $ 1, -frac{3}{4},-frac{3}{4}, frac{9}{4}, -frac{35}{16}$ which satisfies your min-max requirement. The harmonic sum is

$$1 -frac{7}{8} + frac{3}{4} - frac{71}{80} = frac{-1}{80}.$$

More generally, the constraints on $a,b$, $epsilon$ are

$$ 2(a + 2epsilon -1) - b >0$$

(from the max-min restriction), and

$$ epsilon + frac{a}{4}- frac{b}{5} <0 $$

(so we get a counter example). In other words,

$$ 5 epsilon + frac{5a}{4} < b <2a + 4 epsilon -1.$$

For $a> frac{8}{3} + frac{4}{3} epsilon$ these are consistent and leave us room to choose $b$. (And $a$ automatically satisfies our previous requirement that $a>2+2epsilon$.)

We also need $a <4$ and $b<5$ to fit your requirement that $|a_j| le j^alpha$. The requirement $a<4$ only asks that $epsilon <1$, which we already assumed, however $b<5$ forces $epsilon <1/4$. Thus only for $epsilon <1/4$ can you find $a$ and $b$ which make the harmonic sum at order $5$ negative and satisfy all your constraints.