Numberphile’s Proof for the Sum 1+2+3+...

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2022

As we know that Sir Ramanujan gave the solution of sum of all natural numbers up to infinity and said that the sum of all natural numbers till infinity is-1/12. I studied on this topic and found that if we try to solve the infinite series in a slightly different way, then we get the answer of its sum different from-1/12, so this is what I have written in this paper that such Ramanujan Sir, what was the mistake in solving the infinite series, which by solving it in a slightly different way from the same concept, we get different answers.

№2(46) (2018), 2019

Infinities are usually an interesting topic for students, especiallywhen they lead to what seems like paradoxes, when we have two differentseemingly correct answers to the same question. One of such cases is summation of divergent infinite sums: on the one hand, the sum is clearly infinite,on the other hand, reasonable ideas lead to a finite value for this same sum.A usual way to come up with a finite sum for a divergent infinite series is tofind a 1parametric family of series that includes the given series for a specificvalue0of the corresponding parameter and for which the sum convergesfor some other values. For the valuesfor which this sum converges, wefind the expression()for the resulting sum, and then we use the value(0)as the desired sum of the divergent infinite series. To what extent is theresult reasonable depends on how reasonable is the corresponding generalizingfamily. In this paper, we show that from the physical viewpoint, the existingselection of the families is very...

The American Mathematical Monthly

The Mathematics Enthusiast, 2019

This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic series. The whole situation happened directly in the course of the process of a maths lesson. The formulation of the hypothesis was supported by the computer. The hypothesis concerns an interesting connection between harmonic series and the Euler's number. Let denote the 'th partial sum of the harmonic series. Let's notice the sums 1 , 4 , 11 ,. . ., where the partial sum reaches 1, 2, 3,. .. for the first time. Let's mark the relevant indexes 1, 4, 11,. .. as 1 , 2 , 3 ,. . .. So is the index of such a partial sum for which the following is true: −1 < , ≥. The hypothesis lim +1 = has been proved, in an elementary way, in the article.

The concept of an infinite sum is mysterious and intriguing. How can you add up an infinite number of terms? Yet, in some contexts, we are led to the contemplation of an infinite sum quite naturally. For example, consider the calculation of a decimal expansion for 1/3. The long division algorithm generates an endlessly repeating sequence of steps, each of which adds one more 3 to the decimal expansion. We imagine the answer therefore to be an endless string of 3's, which we write .333· · ·. In essence we are defining the decimal expansion of 1/3 as an infinite sum 1/3 = .3 + .03 + .003 + .0003 + · · ·. For another example, in a modification of Zeno's paradox, imagine partitioning a square of side 1 as follows: first draw a diagonal line that cuts the square into two triangular halves, then cut one of the halves in half, then cut one of those halves in half, and so on ad infinitum. (See Figure 1.) Then the area of the square is the sum of the areas of all the pieces, leading to another infinite sum 1 = 1 2 + 1 4 + 1 8 + 1 16 + · · ·. Although these examples illustrate how naturally we are led to the concept of an infinite sum, the subject immediately presents difficult problems. It is easy to describe an infinite series of terms, much more difficult to determine the sum of the series. In this paper I will discuss a single infinite sum, namely, the sum of the squares of the reciprocals of the positive integers. In 1734 Leonhard Euler was the first to determine an exact value for this sum, which had been considered actively for at least 40 years. By today's standards, Euler's proof would be considered unacceptable, but there is no doubt that his result is correct. Logically correct proofs are now known, and indeed, there are many different proofs that use methods from seemingly unrelated areas of mathematics. It is my purpose here to review several of these proofs and a little bit of the mathematics and history associated with the sum.

Mathematics Letters

Four basic problems in Riemann's original paper are found. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s = a+ib). However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis, the two sides of Zeta function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1 but divergent when Re(s) < 1. The integral form of Zeta function does not change the divergence of its series form. Two reasons to cause inconsistency and infinite are analyzed. 3. When the integral form of Zeta function was deduced, a summation formula was used. The applicable condition of this formula is x > 0. At point x = 0, the formula is meaningless. However, the lower limit of Zeta function integral is x = 0, so the formula can not be used. 4. A formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula was also x > 0. However, the lower limit of integral in the deduction was x=0. So this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.

Applicable Algebra in Engineering, Communication and Computing, 2002

A fruitful interaction between a new randomized WZ procedure and other computer algebra programs is illustrated by the computer proof of a series evaluation that originates from a definite integration problem.

Apropos  1 1+ 2 + 3 + 4 + 5 + =-12 Abstract The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative num-bers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1 1 1 1 1 + + + + + and as 1 1 1 1 1 − − − − − − and that the following Equation applies: 1 1 1 1 1 1 1 1 1 1 + + + + + =− − − − − −   The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation 1 1 2 3 4 5 12 + + + + + = −  is discussed in this paper only to the extent that How to cite this paper: Depuydt, L. (2017) Apropos  1 1+ 2 + 3 + 4 + 5 + =-12 :

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