To read my previous note on this click here
What does 1+2+3+4+5+6+... equal?
We need to work with other identities first.
Let S1=1−1+1−1+...;
and S2=1−2+3−4+....
First we solve S1.
S1=1−1+1−...
1−S1=1−(1−1+1−...)
1−S1=1−1+1−...
1−S1=S1
S1=21
Now we solve S2.
S2=1−2+3−4+...
2S2=(1−2+3−4+...)+(1−2+3−4+...)
By shifting the addend one value to the right you get
2S2=1−1+1−1+...
By using our previous value, we get
2S2=21
S2=41
Finally, we solve S
S=1+2+3+4+...
S−S2=(1+2+3+4+...)−(1−2+3−4+...)
S−S2=4+8+12+...
S−S2=4(1+2+3+...)
S−41=4S
S=−121
Ta da!!!
Easy Math Editor
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Comments
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Top NewestWeird...I posted a discussion on these exact sums...
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when, because I want to read it.
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Here it is Interesting sums
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Numberphile!!!!
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No, Wikipedia.
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oh!
I learnt it from that...
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This is a famous result that ζ(−1)=−121. (Saw this in Number Phile :P).
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