Convergence as Closure: A Geometric Reformulation of Infinite Series
Abstract
We present a reformulation of infinite series in which summation is interpreted as a geometric invariant—total variation over a relational manifold—rather than as a sequential limiting process. Convergence is characterized by closure (finite total variation) instead of temporal approach. Standard convergence results are recovered, while providing a structural interpretation that removes implicit temporal assumptions from analysis.
1. Motivation
In standard analysis, an infinite series
$$\sum_{n=1}^{\infty} a_n$$
is defined as the limit of partial sums:
$$S_N = \sum_{n=1}^N a_n, \quad \lim_{N \to \infty} S_N.$$
This formulation implicitly treats the index \( n \) as a sequential parameter. However, nothing in the algebraic structure of \( a_n \) requires \( n \) to represent a process; it may equally be treated as a coordinate labeling sites in a discrete manifold.
2. Discrete Manifold and Magnitude Field
Definition 2.1 (Discrete Relational Manifold)
Let \( \mathcal{M} = \mathbb{N} \) be a one-dimensional discrete manifold with coordinate \( n \in \mathbb{N} \).
Definition 2.2 (Magnitude Field)
A magnitude field is a function
$$M : \mathbb{N} \to \mathbb{R}_{\ge 0},$$
assigning a non-negative magnitude \( M_n \) to each site \( n \).
3. Closure Functional
Definition 3.1 (Total Variation / Closure)
Define the closure functional
$$\Omega(M) := \sum_{n=1}^{\infty} M_n.$$
This quantity is interpreted as the total variation of the magnitude field over the discrete manifold.
4. Convergence as Closure
Theorem 4.1 (Equivalence with Classical Convergence)
Let \( (a_n) \) be a real sequence and define \( M_n = |a_n| \). Then:
$$\sum_{n=1}^{\infty} a_n \text{ converges absolutely} \quad \Longleftrightarrow \quad \Omega(M) < \infty.$$
Proof
By definition of absolute convergence,
$$\sum_{n=1}^{\infty} |a_n| < \infty.$$
This is exactly the finiteness of \( \Omega(M) \). ∎
Corollary 4.2 (Closure Interpretation)
Absolute convergence is equivalent to finite total variation of the magnitude field on \( \mathcal{M} \).
Thus, convergence is a global geometric property, not a sequential process.
5. Power-Law Fields and p-Series
Proposition 5.1 (Power-Law Closure Criterion)
Let
$$M_n = \frac{C}{n^p}, \quad C > 0.$$
Then
$$\Omega(M) < \infty \quad \Longleftrightarrow \quad p > 1.$$
Proof
This is the classical p-series test:
$$\sum_{n=1}^{\infty} \frac{1}{n^p} < \infty \iff p > 1.$$
Reinterpreting the sum as total variation does not alter the condition. ∎
Interpretation
The condition \( p > 1 \) is not about “approaching infinity slowly enough,” but about whether the discrete manifold admits finite closure under the given magnitude decay.
6. Conditional Convergence as Signed Closure Failure
Definition 6.1 (Signed Magnitude Field)
Let
$$a_n = s_n M_n, \quad s_n \in \{+1, -1\}.$$
Proposition 6.2
If
$$\sum_{n=1}^{\infty} M_n = \infty \quad \text{but} \quad \sum_{n=1}^{\infty} a_n \text{ converges},$$
then convergence arises from sign cancellation rather than geometric closure.
Proof
By definition, the unsigned variation diverges while the signed sum converges. Rearrangements alter adjacency relations between signs, destroying cancellation while preserving divergence of total variation (Riemann rearrangement theorem). ∎
Interpretation
Conditional convergence corresponds to non-closed manifolds whose signed projections accidentally cancel. This explains rearrangement instability geometrically.
7. Continuous Limit and Integration
Definition 7.1 (Continuum Magnitude Field)
Let \( M : [1, \infty) \to \mathbb{R}_{\ge 0} \) be measurable.
Define the continuous closure:
$$\Omega(M) := \int_1^{\infty} M(x)\,dx.$$
Proposition 7.2 (Discrete Sampling)
If \( M_n = M(n) \), then
$$\sum_{n=1}^{\infty} M_n$$
is a discrete sampling of the continuous closure integral.
Interpretation
Integration is primary: summation is discrete-coordinate integration. The Riemann integral is not defined here as a limit of sums; rather, sums are approximations of a geometric invariant.
8. Limits Reinterpreted
Proposition 8.1
For a monotone decreasing \( M_n \),
$$\lim_{n \to \infty} a_n = 0$$
is necessary but not sufficient for closure.
Proof
If \( M_n \not\to 0 \), closure diverges trivially. However, \( M_n \to 0 \) does not imply \( \sum M_n < \infty \), as shown by the harmonic series. ∎
Interpretation
Limits describe local asymptotic behavior. Closure describes global structure. Confusing the two is a category error.
9. Reinterpretation of the Riemann Zeta Function
Definition 9.1
For \( p > 1 \),
$$\zeta(p) = \sum_{n=1}^{\infty} \frac{1}{n^p}$$
is the closure invariant of the magnitude field \( M_n = 1/n^p \).
Interpretation
The zeta function parametrizes a family of discrete manifolds by their closure curvature. Divergence at \( p \le 1 \) corresponds to non-closable geometries.
10. Conclusion
We have shown that:
- Infinite series can be interpreted as total variation over discrete manifolds.
- Convergence is equivalent to geometric closure.
- Limits describe asymptotics, not summation.
- Integration is structurally prior to summation.
- Standard analytical results are preserved under reinterpretation.
This framework does not alter computation but clarifies ontology: summation is not a process—it is a geometric invariant.
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