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 \({ S }_{ 1 }\)=1-1+1-1+1-1.......\(\infty\)=1/2

To prove this, first add \({ S }_{ 1 }\) to itself 2\({ S }_{ 1 }\) = 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 2\({ S }_{ 1 }\)=1

**\({ S }_{ 1 }\)=1/2**

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

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

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

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

which would give

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

Hence,

\({ S }_{ 2 }\)=1/4

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

Subtract \({ S }_{ 2 }\) from \({ S }_{ 3 }\)

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

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

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

[typical addition, not as we did above]

We get,

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

```
= 4+8+12+16+20+24........infinite
= 4[1+2+3+4+5+6..........infinite
```

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

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

\({ S }_{ 3 }\) - 1/4 = 4\({ S }_{ 3 }\)

1/4 = 3\({ S }_{ 3 }\)

\(\therefore\) \({ S }_{ 3 }\) = -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!

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

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