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) 8.