Category:P-adic Expansion is a Cauchy Sequence in P-adic Norm
This category contains pages concerning P-adic Expansion is a Cauchy Sequence in P-adic Norm:
Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals numbers $\Q$.
Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.
Then the sequence of partial sums of the series:
- $\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a Cauchy sequence in the valued field $\struct{\Q, \norm{\,\cdot\,}_p}$.
Corollary 1
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic Numbers.
Then the sequence of partial sums of the series:
- $\ds \sum_{n \mathop = m}^\infty d_n p^n$
represents a $p$-adic number of $\struct {\Q_p,\norm {\,\cdot\,}_p}$.
Corollary 2
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Then the sequence of partial sums of the series:
- $\ds \sum_{n \mathop = m}^\infty d_n p^n$
converges to a $p$-adic number in $\struct{\Q_p, \norm{\,\cdot\,}_p}$.
Pages in category "P-adic Expansion is a Cauchy Sequence in P-adic Norm"
The following 3 pages are in this category, out of 3 total.