Series that diverge in R but converge p-adically, partial sum races, and convergence radii
One of the most striking features of the p-adic world is that convergence follows completely different rules than in the real numbers. A series converges p-adically if and only if its terms approach 0 in the p-adic absolute value -- meaning the terms must be divisible by higher and higher powers of p.
This turns familiar intuitions upside down. The geometric series 1 + p + p² + p³ + ... diverges spectacularly in R (each term is bigger than the last!), but converges beautifully in Qp because |pn|p = p−n shrinks to 0. Conversely, the exponential function exp(x), which converges for every real number, has a severely limited radius of convergence in Qp.
The ultrametric inequality gives us a bonus: in Qp, a series converges if and only if its terms tend to 0. There is no need for delicate tests like the ratio or root test -- the "divergence test" becomes both necessary and sufficient.
Watch the geometric series 1 + p + p² + ... diverge to infinity on the real number line while its p-adic terms shrink toward zero. The p-adic sum converges to 1/(1 − p), which is a negative rational number -- a surprising result that perfectly illustrates how p-adic convergence defies real intuition.
The series 1 + p + p² + p³ + ... diverges to infinity in the real numbers, but converges to 1/(1 − p) = -1.0000 in the p-adic numbers. Add terms to watch both behaviors.
| n | Term pn | Sn (real) | |pn|2 |
|---|---|---|---|
| 0 | 1 | 1 | 2-0 = 1.0000 |
Key insight: In the p-adic metric, powers of p are small, not large. The term pn has p-adic absolute value p−n, which goes to 0 as n grows. This is exactly the opposite of real analysis, where pn goes to infinity.
Race real and p-adic partial sums of the factorial series 0! + 1! + 2! + 3! + .... Factorials grow explosively in R, but they contain more and more factors of p, so |n!|p = p−vp(n!) shrinks toward 0. The p-adic valuation vp(n!) counts how many times p divides n!, computed by Legendre's formula: vp(n!) = Σ ⌊n/pk⌋.
The sum of factorials 0! + 1! + 2! + 3! + ... diverges in R (factorials grow fast!) but converges in Qp because vp(n!) grows, making |n!|p shrink toward 0.
| n | n! | v2(n!) | |n!|2 |
|---|---|---|---|
| 0 | 1 | 0 | 1 = 1.0000 |
Key insight: The factorial n! is divisible by increasingly high powers of any prime p. By Legendre's formula, vp(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ..., which grows roughly like n/(p−1). So |n!|p shrinks exponentially, and the factorial series converges in every Qp.
The exponential series exp(x) = Σ xn/n! converges for all real x (infinite radius of convergence), but in Qp it converges only when |x|p < p−1/(p−1). The culprit is the denominator n!: while 1/n! shrinks in R (helping convergence), in Qp we have |1/n!|p = pvp(n!), which grows. This growth fights against |xn|p, and x must be small enough p-adically to win.
The exponential series exp(x) = 1 + x + x²/2! + x³/3! + ... converges for all real x, but in Qp it only converges when |x|p < p−1/(p−1).
For p = 2: radius = 2−1/(2 − 1) = 0.5000. Any x with |x|2 ≥ 0.5000 makes exp(x) diverge p-adically.
| n | |xn/n!| (R) | |xn/n!|2 | v2(n!) |
|---|---|---|---|
| 0 | 1.000000 | 1.000000 | 0 |
| 1 | 1.000000 | 1.000000 | 0 |
| 2 | 0.500000 | 2.000000 (growing!) | 1 |
| 3 | 0.166667 | 2.000000 | 1 |
| 4 | 0.041667 | 8.000000 (growing!) | 3 |
| 5 | 0.008333 | 8.000000 | 3 |
| 6 | 0.001389 | 16.000000 (growing!) | 4 |
| 7 | 0.000198 | 16.000000 | 4 |
| 8 | 2.48e-5 | 128.000000 (growing!) | 7 |
| 9 | 2.76e-6 | 128.000000 | 7 |
| 10 | 2.76e-7 | 256.000000 (growing!) | 8 |
| 11 | 2.51e-8 | 256.000000 | 8 |
| 12 | 2.09e-9 | 1024.000000 (growing!) | 10 |
Key insight: Dividing by n! helps convergence in R (making terms smaller) but hurts convergence in Qp (making p-adic absolute values larger). This is why exp(x) has infinite radius in R but only radius p−1/(p−1) in Qp. For p = 2, this is just 0.5 -- a dramatically smaller domain.