# Tagged Questions

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### $\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
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### Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
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### Proving the sum of the harmonic series up to $p-1$ is divisible by $p$

Wolstenholmes theorem says that $1 + \frac{1}{2} + ... + \frac{1}{p-1} \equiv 0 \bmod p$. I dont quite get the proof, but I was wondering if the argument here is valid for the narrow case that it is ...
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### Finding infinite sequences with pairwise relatively prime outputs.

I am looking for a formula which for every element in $\mathbb{Z}$ as an input, gives pairwise relatively prime outputs. That is for example thanks to Greg Martin's suggestion the positive outputs of ...
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### The Sum $\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$

$$\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$$ Does this sum converge or does it diverge? Are there any results related to this? ($\pi(n)$ is the number of primes less than or equal to $n$)
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### if $F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$ ,then there exist $F_{i}$,such $p|F_{i}$ [duplicate]

Let sequence $\{F_{n}\}$ such $$F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$$ let $p$ is prime number,show that:there is exsit $F_{i}$ such $$p|F_{i},1\le i\le p+1$$ my idea: if $$p=2$$,then then ...
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### sum of infinite roots? [duplicate]

What is the sum of this infinite series of roots: $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4 + \cdots+\sqrt{\infty}}}}}$$ This is an interesting expression because the increase created by the addition of the ...