The Sum of Powers

Recall that in real analysis, we usually say that given the convergence of the series,

$latex\displaystyle 1 + \frac 12 + \frac 14 + \frac 18 + \cdots = 2,$

and

$S := 1 + 2 + 4 + 8 + \cdots = \infty.$

However, if we manipulate the second sum formally,

$2S = 2 + 4 + 8 + \cdots = S - 1,$

yielding the rather suspect claim that

$1 + 2 + 4 + 8 + \cdots = -1.$

We brush this result aside as “absurd” since this series diverges. But what do we mean by diverge? We mean that the partial sums do not converge. What does “converge” mean? It means that terms get “closer” to each other. Well, what does “closer” mean? It turns out that we superimposed our intuition of closeness, that is, given two numbers $x, y$ , we measure their “closeness” by $d(x, y) := |x - y|$ . But what happens if we use a different kind of closeness?

To be absolutely clear, in order to even begin discussing many of our results, we had to construct $\mathbb R$ from $\mathbb Q$ , which we have in fact already done, so that the notion of a metric $d : K \times K \to \mathbb R$ is even well-defined (and useful via the supremum property).

Yet, we already have a useful result that Cauchy sequences in $\mathbb R \equiv \overline{ \mathbb Q }$ converge in $\mathbb R$ . We’re going to use this property to “(Cauchy-)complete” any metric space $(K, d)$ into the metric space $( \overline K, d)$ that contains $K$ as a dense subspace, and setting $K = \mathbb Q$ equipped with a special metric $d$ , the equally following suspect equation actually makes sense:

$\dots 999 := 9 + 90 + 900 + \cdots = -1.$

Henceforth, let $(K, d)$ denote any metric space.

Definition 1. Let $\frak C(K) \subseteq \mathcal F(\mathbb N, K)$ denote the subset of Cauchy sequences in $K$ . We remark that $\frak C(K)$ contains convergent $K$ -sequences as a subset.

Lemma 1. The relation $\sim$ defined on $\frak C(K)$ by $x \sim y$ if and only if $d( x_n, y_n) \to 0$ is an equivalence relation. In particular, for any $\ell \in K$ , $x \sim (\ell, \ell, \ell, \dots)$ if and only if $x_k \to \ell$ .

Corollary 1. Denote $K^* := \frak C(K)/{\sim}$ . The canonical inclusion $\iota : K \to K^*$ defined by $\iota(x) = [(x,x,x,\dots)]$ is a well-defined injection. Henceforth, for $x \in K$ , we denote $x \equiv \iota(x)$ without ambiguity, so that $K \hookrightarrow{} \iota(K) \subseteq K^*$ .

Lemma 2. For any $x, y, z \in K$ ,

$| d(x, y) - d(x, z) | \leq d(y , z) .$

This is a consequence of the reverse triangle inequality for normed spaces.

Lemma 3. The map $d^* : K^* \times K^* \to \mathbb R$ defined by $d^* ([x], [y]) = \lim_{n \to \infty} d(x_n, y_n)$ is a well-defined metric on $K^*$ . Furthermore, $d^* |_{K \times K} = d$ .

Proof. The hardest part of this result is to verify that $d^*$ is well-defined.

Firstly, fix Cauchy $K$ -sequences $x, y$ . Fix $\epsilon > 0$ . Then for $k_1>0, k_2 > 0$ , there exists sufficiently large $N$ such that for $m > n > N$ ,

$d(x_m, x_n) < k_1 \cdot \epsilon, \quad d(y_m, y_n) < k_2 \cdot \epsilon.$

By Lemma 2,

$\begin{aligned} | d(x_m, y_m) - d(x_n, y_n) | &\leq | d(x_m, y_m) - d(x_m, y_n) | + | d(x_m, y_n) - d(x_n, y_n) | \\ & \leq d(y_m, y_n) + d(x_m, x_n) \\ & < (k_1 + k_2) \cdot \epsilon. \end{aligned}$

Setting $k_1 = k_2 = 1/2$ , we conclude that $\{ d(x_n, y_n) \}$ is Cauchy in $\mathbb R$ and hence converges in $\mathbb R$ .

To prove uniqueness, suppose $x \sim x'$ and $y \sim y'$ . Applying bookkeeping on the estimate

$|d(x_n, y_n) - d(x_n', y_n')| \leq d(x_n, x_n') + d(y_n, y_n')$

yields the result.

We leave it as an exercise to verify that $d^*$ is a metric on $K^*$ .

Lemma 4. $\overline{ K } = K^*$ .

Proof. Fix $x \equiv [(x_1, x_2, x_3, \dots)] \in K^*$ . We leave it as an exercise to show that the sequence $\{ x_n^* \}$ converges to $x$ . This means $K^* = \overline{\iota(K)} \cong \overline{K}$ .

Theorem 1. For any metric space $(K, d)$ , there exists a unique complete metric space $(K^*, d^*)$ that contains $(K, d)$ as a subspace. Furthermore, $\overline K = K^*$ .

Proof. Combine Lemmas 1, 3, and 4 to construct $K^*$ with the desired properties. All that remains is to prove that $K^*$ is complete.

Let $\{ y_n \}$ be any Cauchy $K^*$ -sequence. For each $n$ , use Lemma 4 to choose any $x_n \in B_{d^*}( y_n, 1/n ) \cap K$ . We leave it as an exercise to check that $\{ x_n \}$ is Cauchy, and define $y := [ \{x_n\} ]$ . By the triangle inequality, given $n$ ,

$\displaystyle \begin{aligned} d^* (y_n, y) &\leq d^* (y_n, x_n) + d^* (x_n, y) \\ &< \frac 1n + \lim_{k \to \infty} d(x_n, x_k). \end{aligned}$

For sufficiently large $n$ and $k \geq n$ , $1/n < \epsilon/2$ and $d(x_n, x_k) < \epsilon/2$ , yielding $y_n \to y$ .

To resolve the seemingly absurd equation, we will need to define the $p$ -adic metric, which requires a brief discussion on $p$ -adic numbers. Let $p$ be any prime number.

Lemma 5. For any nonzero rational number $r$ , there exists a unique integer $v_r$ and a unique rational number $s_r$ such that

$\displaystyle r = p^{v_r} \cdot s_r.$

Proof. Write $r = (-1)^k \cdot m/n$ for positive integers $m, n$ , and $k = 0$ if and only if $r > 0$ . By the fundamental theorem of arithmetic, find unique positive integers $v_m, q_m, v_n, q_n$ such that

$m = p^{v_m} \cdot q_m, \quad n = p^{v_n} \cdot q_n.$

Defining $v_r := v_m - v_n$ and $s_r := (-1)^k \cdot q_m/q_n$ yields the desired result.

Now we are ready to define the $p$ -adic norm on $\mathbb Q$ , which induces a metric on $\mathbb Q$ .

Lemma 6. The map $| \cdot |_p : \mathbb Q \to \mathbb R$ defined by

$| r |_p := p^{-v_r},$

where $v_r$ is defined according to Lemma 5 if $r \neq 0$ , and $| 0 |_p := 0$ , is a norm on $\mathbb Q$ .

Proof. For non-degeneracy, if $r \neq 0$ , then $v_r \geq 0$ , so that $| r |_p \neq 0$ . For the triangle inequality, we leave it as an exercise to verify that $v_{r + s} \geq \min \{ v_r, v_s \}$ , so that

$\begin{aligned} | r + s |_p &= p^{-v_{r + s}} \\ & \leq p^{ -{ \min \{ v_r , v_s \} } } \\ &= p^{ \max \{ -v_r, - v_s \} } \\ &= \max \{ p^{-v_r}, p^{-v_s} \} \\ &= \max \{ | r |_p , | s |_p \} \\ & \leq | r |_p + | s |_p. \end{aligned}$

Roughly speaking, then, a rational number is small in the $p$ -adic sense if it “contains” a higher power of $p$ .

Example 1. For any power of $2$ , $| 2^k |_2 = 2^{-k}$ .

Definition 2. The set $p$ -adic numbers, denoted by $\mathbb Q_p$ , is defined to be the completion of $\mathbb Q$ under the metric $d_p$ induced by the $p$ –adic norm defined in Lemma 5: $\mathbb Q_p = \overline{\mathbb Q}$ .

Theorem 2. Under the $2$ -adic metric,

$1 + 2 + 4 + 8 + \cdots = -1.$

Proof. Without reservation, the partial sum $s_n$ is given by the geometric sum

$\displaystyle s_n := 1 + 2 + 4 + \cdots + 2^{n-1} = \frac{2^n - 1}{2 - 1} = 2^n - 1,$

so that $s_n - (-1) = 2^n$ . Hence,

$d_2(s_n, -1) = |s_n - (-1)|_2 = | 2^n |_2 = 2^{-n} \to 0.$

Therefore, $s_n \to -1$ , and we get

$1 + 2 + 4 + 8 + \cdots = -1.$

We leave it as an exercise to verify that

$\underbrace{ 9 \dots 9 }_n \to -1,$

when convergence is defined by the $3$ -adic metric. In fact the expression $\dots 999$ , while meaningless in $\mathbb R$ , does have a meaning in $\mathbb Q_p$ , and we can evaluate it to $-1 \in \mathbb Q \subseteq \mathbb Q_p$ .

—Joel Kindiak, 12 Apr 25, 1758H

KindiakMath

The Sum of Powers

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The Sum of Powers

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