Ord function and convergent in Qp

In summary, the conversation discusses solving a problem involving determining the order of a factorial, as well as analyzing the convergence of two series in Qp. The speaker mentions attempting to count the number of terms in (p^n)! that contain a certain factor, and also expresses trouble with analyzing the convergence of one of the series. They ask for assistance with the problem.
  • #1
Funky1981
22
0

Homework Statement


Solve the following :

a) Show that ordp((p^n)!)=1+p+p^2+p^3+...+p^(n-1)

b)For which values of p does the following series converge in Qp?
1)1+(15/7)+(15/7)^2+(15/7)^3+...
2)1!+2!+3!+4!+...


2. The attempt at a solution



For a) I want to to count how many terms of (p^n)! containing the factor p but I failed using my way.

For b) 1) I tried to use the definition of convergent in Qp but when i got the geometric series then it is complicated to analyse p and 2) I have no idea

Can someone help me ?? many thanks
 
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  • #2
Funky1981 said:
For a) I want to to count how many terms of (p^n)! containing the factor p but I failed using my way.
How many terms have the factor p^n? The factor p^(n-1) but not p^n? ...
 

FAQ: Ord function and convergent in Qp

What is the Ord function in Qp?

The Ord function in Qp is a function that assigns a non-negative integer to each element in the field. It represents the number of times the element can be divided by the prime number p without resulting in a fractional value.

What is the significance of the Ord function in Qp?

The Ord function is important in Qp because it helps determine the convergence of a sequence in the field. If the Ord of a sequence approaches infinity, then the sequence converges in Qp.

How is convergence determined in Qp?

In Qp, convergence is determined by the Ord function. A sequence converges if the Ord of the sequence approaches infinity as the index of the sequence increases.

What is the relationship between the Ord function and the p-adic absolute value?

The Ord function and the p-adic absolute value are closely related in Qp. The p-adic absolute value of an element x is equal to p^(-Ord(x)), which represents the "size" of the element in the field.

Can the Ord function be negative in Qp?

No, the Ord function in Qp is always non-negative. This is because the p-adic absolute value of an element x is equal to p^(-Ord(x)), and since p is a positive number, Ord(x) cannot be negative.

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