Open Journal of Mathematics and Physics | Volume 2, Article 75, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/9zm6b | published: 4 Feb 2020 | https://ojmp.wordpress.com CX [microresearch] Diamond Open Access

The infinity theorem

Open Mathematics Collaboration∗† March 19, 2020

Abstract The infinity theorem is presented stating that there is at least one multivalued series that diverge to infinity and converge to infinite finite values.

keywords: multivalued series, infinity theorem, infinite

Introduction

1. 1, 2, 3, ..., ∞ 2. N =x{ N 1, 2,∞3,}...

∞ > ∈ = { } The infinity theorem

3. Theorem: There exists at least one divergent series that diverge to infinity and converge to infinite finite values.

∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Mathematics Collaboration

1 Proof

1 4. S 1 1 1 1 1 1 ... 2 [1] = − + − + − + = (a)

5. Sn 1 1 1 ... 1 has n terms.

6. S = lim+n + S+n +

7. A+ft=er app→ly∞ing the limit in (6), we have S 1 1 1 ...

+ 8. From (4) and (7), S S 2 = 0 + 2 + 0 + 2 ...

1 + 9. S 2 2 1 1 1 .+.. = + + + + + + 1 10. Fro+m (=7) (and+ (9+), S+ )2 2S . + +1 11. Using (6) in (10), limn+ =Sn 2 2 limn Sn. 1 →∞ →∞ 12. limn Sn 2 + = →∞ 1 13. From (6) a=nd (12), S 2. + 1 14. From (7) and (13), S = 1 1 1 ... 2. + = + + + = (b) 15. S 1 1 1 1 1 ...

+ 1 16. S =0 +1 +1 +1 +1 +1 1 ... 2 17. S = S+ 1− 2+ 2− 2+ .−.. + =

+ 1 18. S + 2 =1 +2S+ + + + 1 + 19. S + 2= + + = − 2 (c) 20. S 1 1 1 1 1 ...

+ 1 21. S =0 +0 +1 +1 +1 +1 1 1 ... 2 22. S = S+ 1+ 1− 2+ 2− 2+ .−.. + =

+ 1 23. S + 2 =2 +2S+ + + + + 3 + 24. S + 2= + + = − (n) 2n 1 25. S 2 for n N 1, 2, 3, ... 0 . ( − ) + = − ∈ = { } ∪ { } (n ) 26. Now, we will take n . → ∞ 1 27. (25) can be written a→s S∞ n 2. 28. From (26) and (27), + = − + 1 S lim n . n 2 + = →∞ (− + ) = −∞ 29. From (7) and (26),

n S 1 lim 1 lim n . ∞ n n i 1 i 1 + →∞ →∞ = ∑= = ∑= = = ∞ Final Remarks

30. S converges.

31. S+ diverges.

+ 3 32. In summary, the series S diverges to infinity and converges to

infinite finite values. + 33. The infinite is the quantum superposition of mathematics [2–5].

34. S is a multivalued series.

+ 2n 1 35. S 2 , n N 0 ( − ) + = {− ±∞ ∶ ∈ ∪ { }} A complex multivalued integral

1 x 36. Let S1 0 1 dx.

iθ 37. e co=s θ∫ (i−sin)θ 38. eiπ = 1 +

39. For=n− Z, ei π 2nπ 1. ( + ) 40. Substi∈tuting (39) in=(−36), 1 iπ 2n 1 x S1 e dx. 0 ( + ) 41. = ∫ 1 eiπ 2n 1 x S1 ( + ) iπ 2n 1 0 42. = ∣ eiπ( 2n +1 )1 S1 iπ( 2n+ ) 1 − = 43. Inserting back (39) into the first te(rm +of t)he numerator in (42), 2 S . 1 iπ 2n 1 − = 44. Then, ( + ) 2i S1 n Z . 2n 1 π = { ∶ ∈ } ( +4 ) 45. S is a discrete multivalued sum.

+ 46. S1 is a continuous multivalued sum.

Multivalued subtraction

47. S 1 1 1 ...

48. S+ = 1 + 1+ 1+ 1 ...

49. S+ = 1 + (S + + + )

50. S+ = S+ 1+

51. Si+n−ce S+ =is multivalued, let’s rewrite (50) as S S 1. 52. In (51),+there are the following two cases: + − + = (i) S S , ii S S ( ) + = + . 53. (52i) and+ (≠51)+ lead to 0 1, which is absurd.

54. From (52) and (53), S = S . + + 3 1 55. From (35) and (54), one≠possibility for (51) is S 2, and S 2. + = + = Riemann’s rearrangement theorem

56. [6,7]

57. 1 n 1 1 1 1 1 1 R 1 ... +∞ + n 1 n 2 3 4 5 6 (− ) 58. = ∑= = − + − + − + R ln 2

= 5 59. 1 R ln 2 2 60. = R ...... 61. The positive terms of R sum up to =infinity (R ).

62. The negative terms of R sum up to minus infin→ity∞(R ).

63. Let x R. → −∞

64. From (∈61) and (62), R can be rearranged to

R x a1 a1 a2 a2 a3 a3 ... x.

65. Therefore, = + ( − ) + ( − ) + ( − ) + = R x x R .

= { ∶ ∈ } Open Invitation

Review, add content, and co-author this paper [8,9]. Join the Open Mathematics Collaboration. Send your contribution to [email protected].

Ethical conduct of research

This original work was pre-registered under the OSF Preprints [10], please cite it accordingly [11]. This will ensure that researches are con- ducted with integrity and intellectual honesty at all times and by all means.

6 References

[1] Lobo, Matheus P. “The Convergence of Grandi’s Series.” OSF Preprints, 4 Jan. 2020. https://doi.org/10.31219/osf.io/aq5xz

[2] Lobo, Matheus P. “Superposition as a Mathematical Mapping Between Two Distinct Scales.” OSF Preprints, 31 Dec. 2019. https://doi.org/10.31219/osf.io/uksgx

[3] Lobo, Matheus P. “Quantum Superposition as Entanglement.” OSF Preprints, 25 Dec. 2019. https://doi.org/10.31219/osf.io/m2ajq

[4] Lobo, Matheus P. “Spin Is a Superposition of Circu- lar Charged Trajectories.” OSF Preprints, 30 June 2019. https://doi.org/10.31219/osf.io/tv2rb

[5] Lobo, Matheus P. “Time Travel: Coexistence of Past, Present, and Future?.” OSF Preprints, 2 Sept. 2019. https://doi.org/10.31219/osf.io/7ruay

[6] Wolfram MathWorld. Riemann Series Theorem. Retrieved 2020-3-19. https://mathworld.wolfram.com/RiemannSeriesTheorem.html

[7] StackExchange: Mathematics. Retrieved 2020-3-19. https://math.stackexchange.com/questions/1795509

[8] Lobo, Matheus P. “Microarticles.” OSF Preprints, 28 Oct. 2019. https://doi.org/10.31219/osf.io/ejrct

[9] Lobo, Matheus P. “Simple Guidelines for Authors: Open Jour- nal of Mathematics and Physics.” OSF Preprints, 15 Nov. 2019. https://doi.org/10.31219/osf.io/fk836

[10] COS. Open Science Framework. https://osf.io

[11] Lobo, Matheus P. “The Infinity Theorem.” OSF Preprints, 4 Feb. 2020. https://doi.org/10.31219/osf.io/9zm6b

7 The Open Mathematics Collaboration

Matheus Pereira Lobo (lead author, [email protected]),1,2 Djane da Silva Souza,1 José Carlos de Oliveira Junior1

1Federal University of Tocantins (Brazil); 2Universidade Aberta (UAb, Portugal)

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