1+2+3+4+5........infinity=-1/12, HOW?

This is something which made me amazed and astonished when I saw to it for the first time. Mathematics at its very best, truly! For proving it, we first need to prove that S1{ S }_{ 1 }=1-1+1-1+1-1.......\infty=1/2

To prove this, first add S1{ S }_{ 1 } to itself 2S1{ S }_{ 1 } = 1-1+1-1+1-1........\infty

                          +1-1+1-1+1........infinite

here comes the game changer, we wouldn't add directly to get our answer again to zero, rather, we would add to one no. forward. We would do calculations by neglecting the first no. and then the same. This wouldn't disturb our series as it is an infinite series

Now, we get 2S1{ S }_{ 1 }=1

S1{ S }_{ 1 }=1/2

Here, we would introduce another series S2{ S }_{ 2 }=1-2+3-4+5-6......\infty

Add it to itself 2S2{ S }_{ 2 }=1-2+3-4+5.......\infty

                        +1-2+3-4........infinite      [same type of addition as did above]

\therefore 2S2{ S }_{ 2 }=1-1+1-1+1-1.....\infty

which would give

2S2{ S }_{ 2 }=1/2(proved above S1{ S }_{ 1 }=1/2)

Hence,

S2{ S }_{ 2 }=1/4

Now, S3{ S }_{ 3 } = 1+2+3+4+5.........\infty

Subtract S2{ S }_{ 2 } from S3{ S }_{ 3 }

S3{ S }_{ 3 } - S2{ S }_{ 2 } = 1+2+3+4+5+6.........\infty - [1-2+3-4+5-6.....\infty]

S3{ S }_{ 3 } - S2{ S }_{ 2 } = 1+2+3+4+5+6.........\infty

                                              -1+2-3+4-5+6.............infinite

[typical addition, not as we did above]

We get,

S3{ S }_{ 3 } - S2{ S }_{ 2 } = 0+4+0+8+0+12.........\infty

                                           = 4+8+12+16+20+24........infinite

                                           = 4[1+2+3+4+5+6..........infinite

S3{ S }_{ 3 } - S2{ S }_{ 2 } = 4S3{ S }_{ 3 } [we know that1+2+3..\infty=S3{ S }_{ 3 }]

Put value of S2{ S }_{ 2 }=1/4 proved above in this equation

S3{ S }_{ 3 } - 1/4 = 4S3{ S }_{ 3 }

  • 1/4 = 3S3{ S }_{ 3 }

    \therefore S3{ S }_{ 3 } = -1/12

or

1+2+3+4+5.......\infty = -1/12

Ha, that was something crazy and i got tired in writing that, as well. But mathematics never tire you, only this pc work does...... For the love of Mathematics!

Note by Kartik Sharma
5 years, 4 months ago

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Well, nice work but convergence and divergence would have reduced the work and made it more beautiful.

Kanishk Devgan - 2 years ago

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