## Estimate $\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$

Suppose $$a>1,b>0$$ are real numbers. Consider the summation of the infinite series: $$S=\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$$ How can I give a tight estimation on the summation? Apparently, one can get the upper bound: $$S\le \sum_{k=1} ^\infty \frac 1 {(a+b)^k}=\frac 1 {a+b-1}$$ But it is not tight enough. For example, fix $$a\rightarrow 1$$, and $$b=0.001$$, then $$S=38.969939$$, it seems that $$S=O(\sqrt{1/b})$$. Another example: $$a=1$$, and $$b=0.00001$$,$$S=395.039235$$.

## How to write a formula, $S$, so that the set $P_1, \cdots, P_k$ is not consistent iff $S$ is valid?

A set of propositional formulas $$P_1, \cdots, P_k$$ is consistent iff there is an environment in which they are all true.

Write a formula, $$S$$, so that the set $$P_1, \cdots, P_k$$ is not consistent iff $$S$$ is valid.

## $\bigcap I_\gamma$ if $I_0 \supset \cdots \supset\I_{\gamma} \cdots$ are unbounded

Let be $$\kappa$$ a regular cardinal and $$I_0 \supset \cdots \supset I_{\gamma} \cdots$$ are unbounded subsets of $$\kappa$$ for $$\gamma < \lambda <\kappa$$ where $$\lambda$$ is limit. I want to show $$\bigcap I_\gamma$$ is unbounded. Is it true? It’s easy show it’s not empty for regularity of $$\kappa$$.

## On constructing free action of the cyclic group $\Bbb Z/3\Bbb Z$ on $S^n \times \cdots \times S^n$($n$ is odd)

Can we construct a free action of the cyclic group $$\Bbb Z/3\Bbb Z$$ on $$S^n \times \cdots \times S^n$$($$n$$ is odd) without multiplying any coordinate by $$e^{2\pi i/3}$$? In other words, do I Need to multiply at least one coordinate by a 3rd root of unity to get a free action of $$\Bbb Z/3\Bbb Z$$? I have tried to construct such examples but could not find any.

Thank you so much in advance.

## Hypothesis $\cdots$

for $$n, p\in\mathbb{N}(p-prime\>number)$$ there exist $$m\in\mathbb{N}$$ for $$\p^m\equiv m\pmod n$$