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Let $p$ be a prime number. Can one give an explicit example of a series $\sum_{n\ge0}a_n$ where the $a_n$'s belong to $\mathbb Q$, $\sum_{n\ge0}a_n$ converges in $\mathbb C$ towards a rational $r_1$ and converges towards a rational $r_2$ in $\mathbb Q_p$ with $r_1\ne r_2$?

Thanks in advance

joaopa
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1 Answers1

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We can convert between a rational series and rational sequence by $$b_n=\sum_{k=0}^{n-1} a_k$$ $$a_k=b_{n+1}-b_n$$

Now you can focus on finding just a sequence that converges to two different numbers. A very useful choice is to find a sequence that goes to $0$ in one and $1$ in the other field, here is one such example,

$$b_n = \frac{n!}{1+n!}$$

This goes to $0$ in every p-adic field and $1$ in the reals. So now it's a simple matter of picking any two rational numbers you like, $r_1,r_2$ and now make,

$$c_n = r_1b_n+r_2(1-b_n)$$

Now this converges to your choices.

Merosity
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