Question

sum of all natural number 1+2+3+4+5.................=?

Answer 1

According to Ramanujan's infinte sum, sum of all natural number is -1/12.

Proof :

Let , 

 c = 1 + 2 + 3 + 4 + 5 + 6 +...............  ------- ( 1 )

By multiplying both sides by 4.

⇒ 4 c =  4 + 8 + 12 + 16 + 20 + ........   ---- ( 2 )

By subtracting ( 2 ) from ( 1 ),

⇒ c - 4 c = 1 + 2 - 4 + 3 + 4 - 8 + 5 + 6 - 12 + ........

⇒ - 3 c = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8  ................

The value of ( 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8.......... ) is taken as 1 / 4.

⇒ - 3 c = 1 / 4

∴ c = - 1 / 12.

Answer 2

he infinite series whose terms are thenatural numbers 1 + 2 + 3 + 4 + ⋯ is adivergent series. The nth partial sum of the series is the triangular number

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

which increases without bound as n goes toinfinity. Because the sequence of partial sums fails to converge to a finite limit, the seriesdoes 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, some of which have applications in other fields such as complex analysis[citation needed], quantum field theory[citation needed], and string theory. Many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization andRamanujan summation assign the series a value of − 1/12, which is expressed by a famous formula:[2]

{\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.

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