Summation (Again!!!)

To read my previous note on this click here

What does 1+2+3+4+5+6+...1 + 2 + 3 + 4 + 5 + 6 + ... equal?

We need to work with other identities first.

Let S1=11+11+...S_1 = 1 - 1 + 1 - 1 + ...;

and S2=12+34+...S_2 = 1 - 2 + 3 - 4 + ....

First we solve S1S_1.

S1=11+1...S_1 = 1 - 1 + 1 - ...

1S1=1(11+1...)1 - S_1 = 1 - (1 - 1 + 1 - ...)

1S1=11+1...1 - S_1 = 1 - 1 + 1 - ...

1S1=S11 - S_1 = S_1

S1=12S_1 = \frac {1}{2}

Now we solve S2S_2.

S2=12+34+...S_2 = 1 - 2 + 3 - 4 + ...

2S2=(12+34+...)+(12+34+...)2S_2 = (1 - 2 + 3 - 4 + ...) + (1 - 2 + 3 - 4 + ...)

By shifting the addend one value to the right you get

2S2=11+11+...2S_2 = 1 - 1 + 1 - 1 + ...

By using our previous value, we get

2S2=122S_2 =\frac {1}{2}

S2=14S_2 = \frac {1}{4}

Finally, we solve SS

S=1+2+3+4+...S = 1 + 2 + 3 + 4 + ...

SS2=(1+2+3+4+...)(12+34+...)S - S_2 = (1 + 2 + 3 + 4 + ...) - (1 - 2 + 3 - 4 + ...)

SS2=4+8+12+...S - S_2 = 4 + 8 + 12 + ...

SS2=4(1+2+3+...)S - S_2 = 4 (1 + 2 + 3 + ...)

S14=4SS - \frac {1}{4} = 4S

S=112S = -\frac {1}{12}

Ta da!!!

Note by Sharky Kesa
5 years, 9 months ago

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1 vote

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Comments

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Weird...I posted a discussion on these exact sums...

Bogdan Simeonov - 5 years, 9 months ago

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when, because I want to read it.

Sharky Kesa - 5 years, 9 months ago

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Here it is Interesting sums

Bogdan Simeonov - 5 years, 8 months ago

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@Bogdan Simeonov Wow. I found this when I was surfing the internet.

Sharky Kesa - 5 years, 8 months ago

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Numberphile!!!!

Kartik Sharma - 5 years, 5 months ago

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No, Wikipedia.

Sharky Kesa - 5 years, 5 months ago

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oh!

I learnt it from that...

Kartik Sharma - 5 years, 5 months ago

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This is a famous result that ζ(1)=112\zeta (-1) = -\dfrac{1}{12}. (Saw this in Number Phile :P).

Surya Prakash - 3 years, 11 months ago

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