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If $S=\{U_1,U_2,U_3 \dots\}$ is the set of all finite subsets of the positive integers, does the following converge? $$\sum_{i=1}^{\infty}\frac{1}{\sum\limits_{a\in U_i}a^a}$$

Does the following diverge for all $n$? $$\sum_{i=1}^{\infty}\frac1{\sum\limits_{a\in U_i}a^n}$$

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up vote 3 down vote accepted

Consider $\mathcal U_m$ the set of the finite subsets of $\mathbb N$ which admit $m$ as a maximum. Obviously, every non empty finite subset of $\mathbb N$ belongs to one and only one $\mathcal U_m$. Moreover, the cardinal of $\mathcal U_m$ is $2^m$.

Consider $f:\mathbb N \to \mathbb N$ a non decreasing function.

It is clear that if $U$ is in $\mathcal U_m$, then $$ f(m) \leq \sum_{a\in U} f(a) \leq (m+1)f(m) $$ Thus $$ \frac{2^m}{(m+1)f(m)} \leq \sum_{U\in \mathcal U_m} \frac{1}{\sum_{a\in U} f(a)} \leq \frac{2^m}{f(m)} $$

Now, set $f(a) = a^n$ or $f(a) = a^a$ and you have your answer by summing over $m$.

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With the convention that $0$ belongs to $\mathbb N$. –  Did Feb 7 '12 at 11:41
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