Is the power series convergent in the field of p-adic numbers?

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    2015
In summary, the concept of convergence in p-adic numbers differs from that in real numbers. A power series is considered convergent in p-adic numbers if its terms become arbitrarily small in terms of powers of the prime number p as the index approaches infinity. To determine convergence in p-adic numbers, one can use Hensel's lemma or the p-adic absolute value. A power series can be convergent in both real numbers and p-adic numbers, but the convergence behavior may vary. The relationship between convergence in real numbers and p-adic numbers is complex, but in some cases, convergence in one field implies convergence in the other. P-adic numbers and their convergence have applications in various fields such as number theory, cryptography
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Euge
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Here is this week's POTW:

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Find the domain of convergence of the power series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}x^n}{n}$$ in the field $\Bbb Q_p$ of $p$-adic numbers.

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  • #2
No one answered this week's problem. Here is my solution below.

Since $(|(-1)^{n+1}/n|_p)^{1/n} = (|1/n|_p)^{1/n} = p^{\frac{\text{ord}_p n}{n}} \to 1$ as $n\to \infty$, by the ratio test, the series converges for $|x|_p < 1$ and diverges for $|x|_p > 1$. When $|x|_p = 1$, $|(-1)^{n+1}x^n/n|_p = p^{\text{ord}_p n} \ge 1$ for all $n$, which implies that the series diverges. Hence, the domain of convergence is the open unit disk in $\Bbb Q_p$ centered at $0$, or alternatively the closed disk in $\Bbb Q_p$ centered at $0$ of radius $1/p$.
 

FAQ: Is the power series convergent in the field of p-adic numbers?

1. Is the concept of convergence the same in p-adic numbers as in real numbers?

No, the concept of convergence in p-adic numbers is different from that in real numbers. In real numbers, a series is considered convergent if the terms approach zero as the index approaches infinity. In p-adic numbers, a series is considered convergent if the terms become arbitrarily small in terms of powers of the prime number p as the index approaches infinity.

2. How do we determine if a power series is convergent in the field of p-adic numbers?

There are several ways to determine convergence in p-adic numbers. One method is to use Hensel's lemma, which states that a power series is convergent if its coefficients satisfy certain congruence conditions with respect to the prime number p. Another method is to use the p-adic absolute value, which measures the size of a number in terms of powers of p. A power series is convergent if the absolute value of its terms approaches zero as the index approaches infinity.

3. Can a power series be convergent in both real numbers and p-adic numbers?

Yes, it is possible for a power series to be convergent in both real numbers and p-adic numbers. However, the convergence behavior may be different in each field. A series may converge in real numbers but not in p-adic numbers, or vice versa.

4. What is the relationship between the convergence of a power series in real numbers and its convergence in p-adic numbers?

The relationship between convergence in real numbers and p-adic numbers is complex. In general, a power series may converge in one field but not in the other. However, there are some cases where convergence in one field implies convergence in the other. For example, if a power series is absolutely convergent in real numbers, then it is also convergent in p-adic numbers.

5. Are there any applications of p-adic numbers and their convergence to real-world problems?

Yes, p-adic numbers and their convergence have applications in various fields such as number theory, cryptography, and physics. In number theory, p-adic numbers are used to study Diophantine equations and other algebraic structures. In cryptography, they are used for secure communication and data encryption. In physics, p-adic numbers and their convergence are used in quantum mechanics and string theory.

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