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Extra info for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

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This ends our discussion of elementary p-adic analysis. We have not touched many important topics such as differentiability, integration and so on. For these I suggest you look at Koblitz [5]. I particularly recommend his discussion of p-adic integration, Γ-function and ζ-function. The world of p-adic analysis is in many ways very similar to that of classical real analysis, but it is also startlingly different. I hope you have enjoyed this sampler. We will now move on to consider something more like the complex numbers in the p-adic context.

29 This has norm (n + 1)pn+1 p = |n + 1|p pn+1 p 1 , pn+1 which clearly tends to 0 as n → ∞ in the real numbers. 1, (sn ) is a Cauchy sequence and therefore it has a limit in Qp . In real analysis, there are series which converge but are not absolutely convergent. For ∑ ∑ example, the series (−1)n /n converges to − ln 2 but 1/n diverges. Our next result shows that this cannot happen in Qp . 4. The series αn in Qp converges if and only if (αn ) is a null sequence. ∑ Proof. 1 the sequence of partial sums (sn ) is Cauchy since sn+1 − sn = αn is a null sequence.

For n 1, let X(X − 1) · · · (X − n + 1) n! and C0 (X) = 1; in particular, for a natural number x, ( ) x Cn (x) = . n Cn (X) = (a) Show that if x ∈ Z then Cn (x) ∈ Z. (b) Show that if x ∈ Zp then Cn (x) ∈ Zp . (c) If αn ∈ Qp , show that the series ∞ ∑ αn Cn (x), n=0 converges for all x ∈ Zp if and only if lim αn = 0. n→∞ ∑ n (d) For x ∈ Z, determine ∞ n=0 Cn (x)p . ∑ ∑ 3-11. (a) Let αn be a series in Qp . , αn converges if αn+1 λ = lim n−→∞ αn p exists and λ < 1. , if γn λ = lim n→∞ γn+1 p ∑ exists then γn X n converges if |x|p < λ and diverges if |x|p > λ.

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