Abstract
One word, "diverges", names at least four different states. This paper separates them in numbers and shows that the last is relative to the choice of metric. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, bounded oscillation. The partial sums of $1-1+1-1+\cdots$ run $1,0,1,0$, and the Cesàro average gives $0.500000000000$ while the Abel sum reaches $0.50000025$: the two agree (Section 2).
Second, unbounded oscillation. For $1-2+3-4+\cdots$, $(C,1)$ sits at $0.500025001$ at odd $n$ and at $0$ at even $n$ and never settles — $(C,2)$ gives $0.250139673$ and Abel $0.250025001875$, both $1/4$ (Section 3).
Third, radius of convergence zero. At $x=0.1$ the Euler series approaches $0.09158192$ by the eleventh term and flies out to $-0.0195940911$ by the twenty-sixth — and its Borel sum $0.091563333939788081876$ satisfies the original differential equation to a residual of $8.11\times10^{-19}$ (Section 4).
Fourth, and this is the core. The partial sums of $1+2+4+8+\cdots$ run away from $-1$ in the real metric, $4.0\to2.048\times10^{3}\to2.199\times10^{12}$, and converge to $-1$ in the $2$-adic metric, $0.25\to4.88\times10^{-4}\to4.55\times10^{-13}$ (Section 5).
Fifth, and this is the fence. On $1+2+3+\cdots$, Abel gives $999000$, $(C,1)$ gives $6.6677\times10^{7}$ and $(C,2)$ gives $2.2229\times10^{7}$: all three run to infinity — so $-1/12$ is not what any summation method returns (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — Cesàro summation, Abel summation, Borel summation, Frobenius's theorem, the $p$-adic distance, the Euler series as a formal solution, and the divergence of perturbation series are all standard. No general theory of summation is built: regularity, linearity and stability are quoted and not proved. No Tauberian theorem is treated; the converse implications are not entered. No $p$-adic analysis is entered; the $2$-adic distance appears only as a witness. No history is written, and no claim is made that drawing a distinction is what solved anything — such stories always look that way in hindsight, and this paper offers a census only, with no claim that a census solves anything. Quantum field theory is not discussed; the divergence of perturbation series is left to the references. Relation to earlier papers: Paper 211 showed that an asymptotic expansion is best at the $1/x$-th term; this paper takes the other road, not truncation but summation. Paper 232 placed $1+2+3+\cdots=-1/12$ as a convention reached by analytic continuation; this paper supplies the check, showing in numbers that three summation methods actually fail on it. Paper 234 counted the ultrametric; the $2$-adic distance of Section 5 uses that tool directly. Paper 197 counted four distinct roots of "converges"; this is its mirror image. What is added is the four divergences set out in one format, the two-sided measurement showing $(C,1)$ holds two values by parity, the residual check that the Borel sum satisfies the original equation, the distance of one sequence to $-1$ under two metrics side by side, and the numerical demonstration that three summation methods diverge on $1+2+3+\cdots$.
1. Notation and Position
Definition 1.1 a series $\sum a_n$ diverges when the sequence of partial sums $s_n$ does not converge.
Two things are not written in that definition — in which metric convergence is measured, and whether anything besides partial sums is allowed. This paper varies both.
The three conditions on a summation method (quoted here, not proved):
- Regularity: on a convergent series it returns the original value.
- Linearity: it respects sums and scalar multiples.
- Stability: removing the first term changes the value by exactly that term.
Because of these three, arbitrary values cannot be assigned. Section 6 is the demonstration.
Position: Paper 211 treated where to truncate; this paper takes the side that does not truncate but admits another operation.
2. The First Kind: Bounded Oscillation
Computation 2.1: partial sums and Cesàro averages of $1-1+1-1+\cdots$.
| terms | partial sum | Cesàro average |
|---|---|---|
| $1$ | $1$ | $1.000000000000$ |
| $2$ | $0$ | $0.500000000000$ |
| $3$ | $1$ | $0.666666666667$ |
| $10$ | $0$ | $0.500000000000$ |
| $20000$ | $0$ | $\mathbf{0.500000000000}$ |
Computation 2.2: the Abel sum, the limit of $1/(1+x)$.
| $x$ | $1/(1+x)$ |
|---|---|
| $0.9$ | $0.526315789474$ |
| $0.99$ | $0.502512562814$ |
| $0.999$ | $0.500250125063$ |
| $0.999999$ | $\mathbf{0.50000025}$ |
Observation 2.3 the two methods returned the same $1/2$. Not by accident — Cesàro summable implies Abel summable with the same value (Frobenius). This paper quotes and does not prove it.
3. The Second Kind: Unbounded Oscillation
Computation 3.1: measure $1-2+3-4+\cdots$.
| terms | partial sum | $(C,1)$ | $(C,2)$ |
|---|---|---|---|
| $9$ | $5$ | $0.555555556$ | $0.377072310$ |
| $10$ | $-5$ | $0.000000000$ | $0.339365079$ |
| $999$ | $500$ | $0.500500501$ | $0.252296826$ |
| $1000$ | $-500$ | $0.000000000$ | $0.252044530$ |
| $19999$ | $10000$ | $0.500025001$ | $0.250152181$ |
| $20000$ | $-10000$ | $\mathbf{0.000000000}$ | $\mathbf{0.250139673}$ |
$(C,1)$ holds $1/2$ at odd $n$ and $0$ at even $n$ forever, and therefore does not converge. (QED)
Computation 3.2: the Abel sum, the limit of $1/(1+x)^2$.
| $x$ | $1/(1+x)^2$ |
|---|---|
| $0.9$ | $0.277008310249$ |
| $0.99$ | $0.252518875786$ |
| $0.9999$ | $\mathbf{0.250025001875}$ |
Observation 3.3 one series is not summable by one method and equal to $1/4$ by two others. "A divergent series has a sum" means nothing until the method is named — in the language of Paper 16, outside its fence it is empty.
Observation 3.4 (a trap avoided here) $(C,1)$ was first tabulated at even $n$ only. The column then reads $0.000000000$ throughout, which looks like convergence to $0$ — an oscillating quantity must be read at both phases. The same kind of trap as the central difference of an even function in Paper 238.
4. The Third Kind: Radius of Convergence Zero
Computation 4.1: sum the Euler series $\sum_{n\ge0}(-1)^n n!\,x^{n+1}$, the formal solution of $x^2y'+y=x$, at $x=0.1$.
| terms | partial sum | size of the last term |
|---|---|---|
| $1$ | $0.1$ | $0.1$ |
| $6$ | $0.09152$ | $0.00012$ |
| $11$ | $\mathbf{0.09158192}$ | $3.6288\times10^{-5}$ |
| $16$ | $0.0914840440064$ | $0.00013076744$ |
| $21$ | $0.09319427285184512$ | $0.002432902$ |
| $26$ | $\mathbf{-0.01959409108642064}$ | $0.1551121$ |
The series diverges at every $x\ne0$, the terms growing like $n!$. (QED)
Computation 4.2: compute the Borel sum $x\int_0^\infty e^{-t}/(1+xt)\,dt$ and put it back into the equation.
| quantity | value |
|---|---|
| the Borel sum | $0.091563333939788081876$ |
| residual of $x^2y'+y-x$ | $\mathbf{8.11\times10^{-19}}$ |
Observation 4.3 a divergent series acquired one meaningful value, and that value actually satisfies the equation the series was the formal solution of — "divergent, therefore meaningless" is false. Where best to truncate was the subject of Paper 211; this is the road that does not truncate.
5. The Core: Divergence Is Relative to a Metric
Computation 5.1: the partial sums of $1+2+4+8+\cdots$ are $2^{n+1}-1$. Measure their distance to $-1$ in two metrics.
| $n$ | partial sum | real distance | $2$-adic distance |
|---|---|---|---|
| $1$ | $3$ | $4.000$ | $0.25$ |
| $3$ | $15$ | $1.600\times10^{1}$ | $0.0625$ |
| $10$ | $2047$ | $2.048\times10^{3}$ | $4.883\times10^{-4}$ |
| $20$ | $2097151$ | $2.097\times10^{6}$ | $4.768\times10^{-7}$ |
| $40$ | $2199023255551$ | $\mathbf{2.199\times10^{12}}$ | $\mathbf{4.547\times10^{-13}}$ |
Observation 5.2 one and the same sequence runs away under one metric and converges geometrically to $-1$ under the other. In the $2$-adic metric $|s_n-(-1)|_2=2^{-(n+1)}$ exactly — so the sentence "this series diverges" has the fence "in which metric" omitted.
Observation 5.3 the $2$-adic distance used here is the ultrametric counted in Paper 234. There it made every triangle isosceles — here it turns divergence into convergence. This paper does not enter $p$-adic analysis; the distance appears only as a witness.
Observation 5.4 and both answers are correct. Writing $-1$ on the real side is false, and writing "divergent" on the $2$-adic side is false — the two are not competing. They answer different questions.
6. The Fence: Arbitrary Values Cannot Be Assigned
Computation 6.1: apply the methods of Sections 2 and 3 to $1+2+3+\cdots$.
| terms | partial sum | $(C,1)$ | $(C,2)$ |
|---|---|---|---|
| $10$ | $5.5000\times10^{1}$ | $2.2000\times10^{1}$ | $9.5000$ |
| $100$ | $5.0500\times10^{3}$ | $1.7170\times10^{3}$ | $5.8950\times10^{2}$ |
| $1000$ | $5.0050\times10^{5}$ | $1.6717\times10^{5}$ | $5.5890\times10^{4}$ |
| $20000$ | $2.0001\times10^{8}$ | $\mathbf{6.6677\times10^{7}}$ | $\mathbf{2.2229\times10^{7}}$ |
Computation 6.2: the Abel sum, $x/(1-x)^2$.
| $x$ | $x/(1-x)^2$ |
|---|---|
| $0.9$ | $90$ |
| $0.99$ | $9900$ |
| $0.999$ | $\mathbf{999000}$ |
Observation 6.3 all three run to infinity. So $-1/12$ is not what any of these methods returns — "one may assign a divergent series whatever value one likes" is false.
Observation 6.4 $-1/12$ arrives from a different mechanism, analytic continuation — which is what Paper 232 placed as a convention. What this paper adds is the number showing that the summation side genuinely fails.
7. Separator and Fence
Observation 7.1 (the census)
| kind of divergence | example | what gives it a value |
|---|---|---|
| bounded oscillation | $1-1+1-1$ | an average (Cesàro, Abel: $1/2$) |
| unbounded oscillation | $1-2+3-4$ | two averages ($(C,2)$, Abel: $1/4$) |
| radius of convergence zero | the Euler series | an integral transform (Borel) |
| divergent in the real metric | $1+2+4+8$ | another metric ($2$-adic: $-1$) |
| none of them work | $1+2+3+\cdots$ | no summation method does |
Observation 7.2 (the separator) there are two — which metric measures it, and what is allowed besides partial sums (averages, weights, integral transforms). The symbol $\sum a_n$ writes neither — the same shape as $f'$ in Paper 233, $d(x,y)$ in Paper 234, $-u''$ in Paper 235 and "= 0" in Paper 236.
Observation 7.3 (the fence) there are three. First, regularity, linearity and stability, which is why arbitrary values cannot be assigned (Section 6). Second, by which method: $1-2+3-4$ is not summable by $(C,1)$ and equals $1/4$ by $(C,2)$ (Section 3). Third, in which metric: $1+2+4+8$ diverges in the reals and converges $2$-adically (Section 5).
Observation 7.4 (what this paper does not do) no claim is made that drawing a distinction is what solved anything. Such stories always look that way in hindsight — what was done here is a census, and no claim is made that a census solves anything.
8. Census and Closing
What this paper measured.
- Cesàro and Abel agreeing (2.1, 2.2): $0.500000000000$ and $0.50000025$.
- $(C,1)$ holding $0$ and $1/2$ by parity (3.1): no convergence.
- The Borel sum satisfying the equation (4.2): residual $8.11\times10^{-19}$.
- One sequence pointing opposite ways in two metrics (5.1): $2.199\times10^{12}$ against $4.547\times10^{-13}$.
- Three summation methods failing on $1+2+3+\cdots$ (6.1, 6.2).
References
- Euler, L. (1760). De seriebus divergentibus. Novi Commentarii academiae scientiarum Petropolitanae, 5.
- Abel, N. H. (1881). Œuvres complètes de Niels Henrik Abel, tome 2. Christiania: Grøndahl.
- Cesàro, E. (1890). Sur la multiplication des séries. Bulletin des Sciences Mathématiques, 14.
- Borel, É. (1899). Mémoire sur les séries divergentes. Annales scientifiques de l'École Normale Supérieure, 16.
- Hensel, K. (1897). Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 6.
- Hardy, G. H. (1949). Divergent Series. Oxford: Clarendon Press.
- Dyson, F. J. (1952). Divergence of perturbation theory in quantum electrodynamics. Physical Review, 85.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). “Converges” Is Four Separate Claims, and What Separates Them Is What You May Discard [Paper 197]. Zenodo. 10.5281/zenodo.22107590
- Yamagishi, Y. (2026). Adding More Terms Takes You Further from the Answer [Paper 211]. Zenodo. 10.5281/zenodo.22111772
- Yamagishi, Y. (2026). Learn from a Book with the Proofs Removed and the Equals Sign Loses Its Type [Paper 232]. Zenodo. 10.5281/zenodo.22127528
- Yamagishi, Y. (2026). “Distance” Is Six Things [Paper 234]. Zenodo. 10.5281/zenodo.22128226
- Yamagishi, Y. (2026). “Regular” Is Three Things [Paper 238]. Zenodo. 10.5281/zenodo.22129257
Reference
Yamagishi, Y. (2026). “Diverges” Is Four Things, and One of Them Converges in Another Metric [Paper 239]. YOI: https://www.yamagishiyuuki.com/yoi/239
DOI: 10.5281/zenodo.22129461
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 239. This paper is a preprint and has not been peer reviewed.