Ameha Tefera Tessema
Commercial Bank of Ethiopia, Addis Ababa, Ethiopia
Correspondence to: Ameha Tefera Tessema, Commercial Bank of Ethiopia, Addis Ababa, Ethiopia.
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Copyright © 2021 The Author(s). Published by Scientific & Academic Publishing.
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Abstract
Let such that then Where p=2t+1 for all p odd Natural numbers(N), p=2t+2 for all p even natural numbers(N).
Keywords:
Power series, Formula
Cite this paper: Ameha Tefera Tessema, A P-Series Formula, Applied Mathematics, Vol. 11 No. 2, 2021, pp. 21-22. doi: 10.5923/j.am.20211102.02.
1. Introduction
In the early time fermat, a French number theorist, had stated that the equation has no solution in integers, if n>2 [1]. The problem to find the exact value of z is enhanced as the number of terms increases rather than power two. Let us have n terms each of which has a common power p in the series then the series is called a p-series which is applied by the following theorem for p>2 which is also an answer for the claim of fermat.Theorem 1. Let such that thenWhere p=2t+1 for all p odd Natural numbers(N), p=2t+2 for all p even natural numbers.Proof. Let such that if where then multiplying up to n times we get , therefor we can write this expression to p power as If the binomial theorem for expansion .Can be written as Where , Putting and dividing both sides by we get | (1) |
The other way of writing is which can be written to p power as If binomial theorem for expansion can be written as Where ,Putting in and dividing both sides by we get | (2) |
Therefor, the summation of EQ(1) and EQ(2) can be put as followWhere p=2t+1 for all p odd natural numbers, whereas p=2t+2 for all p even natural numbersLet put and if be divided by such that (Real numbers) then we get the following resultThis result led us to a conclusion that every couple terms which have common power can undergo this formula.
References
[1] | Gerhard Frey, The way to the proof of Fermat’s Last Theorem, Annales de la faculte des sciences de Toulouse, Vol. XVIII, 2009, pp.5-23. |