P-adic metric calculate limit

[Lambda96]

Homework Statement

Calculate the following limits ##\displaystyle{\lim_{n \to \infty}} p^n## and ##\sum\limits_{k=0}^{\infty} p^k##

Relevant Equations

none

Hi,

I'm not sure if I have calculated the task here correctly

Task 4-4b looked like this

I have now obtained the following with ##n=-v_p(x-y)##

$$\displaystyle{\lim_{n \to \infty}} p^n= \infty$$
$$\sum\limits_{n=0}^{\infty} p^n=\frac{p}{p-1}$$

Is that correct?

 

[fresh_42]

Homework Statement: Calculate the following limits ##\displaystyle{\lim_{n \to \infty}} p^n## and ##\sum\limits_{k=0}^{\infty} p^k##
Relevant Equations: none

Hi,

I'm not sure if I have calculated the task here correctly

View attachment 345019
Task 4-4b looked like this
View attachment 345020

I have now obtained the following with ##n=-v_p(x-y)##

$$\displaystyle{\lim_{n \to \infty}} p^n= \infty$$
$$\sum\limits_{n=0}^{\infty} p^n=\frac{p}{p-1}$$

Is that correct?

a) What is the distance between ##p^n## and ##0##?

b) What is ##\displaystyle{\left(\sum_{k=0}^\infty p^k \right)\cdot (1-p)}##?

Here is another short introduction to p-adic numbers:
https://www.physicsforums.com/insig...l-number-systems-that-we-have/#p-adic-Numbers

 

[Lambda96]

Thank you fresh_42 for your help

Regarding a)
Isn't the distance calculated as follows? ##| n-0 |_p=| p^n |_p=p^{-n}##

Regarding b)
##\sum\limits_{k=0}^{\infty} p^k \cdot (1-p)=1##

 

[fresh_42]

Thank you fresh_42 for your help

Regarding a)
Isn't the distance calculated as follows? ##| n-0 |_p=| p^n |_p=p^{-n}##

Regarding b)
##\sum\limits_{k=0}^{\infty} p^k \cdot (1-p)=1##

a) Yes. And if ##n\to \infty ## then ##\displaystyle{\lim_{n \to \infty}|p^{n}|_p=\lim_{n \to \infty}p^{-n}}=?##
b) Yes.

Convergence of series is easier in p-adic analysis:
$$
\lim_{n \to \infty}a_np^n=0 \Longrightarrow \sum_{n=k}^{\infty}a_np^n < \infty \;(k\in \mathbb{Z}\, , \,a_n\in \{0,1,\ldots,p-1\})
$$
and all elements of ##\mathbb{Q}_p ## are a limit of such a series.

 

[Lambda96]

Thanks again for your help fresh_42


The limit value of ##\displaystyle{\lim_{n \to \infty}} p^{-n}## should then be ##\displaystyle{\lim_{n \to \infty}} p^{-n}=0##

 

[fresh_42]

Thanks again for your help fresh_42


The limit value of ##\displaystyle{\lim_{n \to \infty}} p^{-n}## should then be ##\displaystyle{\lim_{n \to \infty}} p^{-n}=0##

Yes. The distance between ##p^n## and zero in the p-adic metric goes to zero.

 

[WWGD]

Yes. The distance between ##p^n## and zero in the p-adic metric goes to zero.

Can you remind me of a result to the effect that a property that holds for all p-adics ; p=2,3,... also holds for the Reals? Can't remember the qualifications.

 

[fresh_42]

Maybe somewhat more formal, or to highlight using theusing the p-adic valuation : ##Lim_{n \rightarrow \infty} \frac{1}{p^n} ##

All I could find was

Helmut Hasse (1898 - 1979) showed in his dissertation 1921 about quadratic forms that rational equations can be solved - up to many complicated technical details - if they can be solved for real numbers and all p-adic numbers.

I have a book about quadratic forms that deals with this kind of algebraic topology. But even the language needs an introduction, or do you know by heart what a (prime) spot is, or class numbers? It's all so long ago.