Math, asked by Anonymous, 1 year ago

can anyone prove sum of all natural num is -1/12​

Answers

Answered by Princess1234567
2

Heya....❤❤❤

here is ur answer...

The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number

{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},} \sum_{k=1}^n k = \frac{n(n+1)}{2},

which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.

Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −

1

/

12

, which is expressed by a famous formula,

{\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},} {\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}

where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.

In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".

hope it helps......

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bt not sure abt the answer.....:-( :-( :-(

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Answered by Shubhendu8898
9

Question:- Prove  that sum of n natural number  is  n(n+1)/2

Answer: n(n+1)/2

Step-by-step explanation:

Natural numbers  are,

1 , 2 , 3 , 4, .......................n-1 , n

This is  an AP

First  term(a)  = 1

Last  term(l)  = n

Number  of terms(n) = n

We know that,

Sum  of n terms  of an AP having first term a and last term l is,

=n(n + l)/2

=n(n + 1)/2

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