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Question Number 108609 by Rio Michael last updated on 18/Aug/20

1 + 2 + 3 + 4 + 5 + ... ∞ = ???

$$\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:\mathrm{5}\:+\:...\:\infty\:=\:??? \\ $$

Commented by ajfour last updated on 18/Aug/20

S_n =((n(n+1))/2)

$${S}_{{n}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$

Answered by Dwaipayan Shikari last updated on 18/Aug/20

S=1+2+3+4+5+6+7+....  4S=    4    +    8   +    12   −3S=1−2+3−4+5−6+....  −3S=(1/4)  S=−(1/(12))  ⊛  S=1+2+3+4+5+6+7+8+9+...  S= 1+(2+3+4)+(5+6+7)+(8+9+10)+...  S=1+9+18+27+....  S=1+9(1+2+3+4+....)  S=1+9S  S=−(1/8)  ⊛S=1+(1+1)+(1+1+1)+....      S=          1         +(1+1)+.....  ......  subtracting  0=1+1+1+1+1+....

$${S}=\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+.... \\ $$$$\mathrm{4}{S}=\:\:\:\:\mathrm{4}\:\:\:\:+\:\:\:\:\mathrm{8}\:\:\:+\:\:\:\:\mathrm{12}\: \\ $$$$−\mathrm{3}{S}=\mathrm{1}−\mathrm{2}+\mathrm{3}−\mathrm{4}+\mathrm{5}−\mathrm{6}+.... \\ $$$$−\mathrm{3}{S}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${S}=−\frac{\mathrm{1}}{\mathrm{12}} \\ $$$$\circledast \\ $$$${S}=\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}+... \\ $$$${S}=\:\mathrm{1}+\left(\mathrm{2}+\mathrm{3}+\mathrm{4}\right)+\left(\mathrm{5}+\mathrm{6}+\mathrm{7}\right)+\left(\mathrm{8}+\mathrm{9}+\mathrm{10}\right)+... \\ $$$${S}=\mathrm{1}+\mathrm{9}+\mathrm{18}+\mathrm{27}+.... \\ $$$${S}=\mathrm{1}+\mathrm{9}\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+....\right) \\ $$$${S}=\mathrm{1}+\mathrm{9}{S} \\ $$$${S}=−\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\circledast{S}=\mathrm{1}+\left(\mathrm{1}+\mathrm{1}\right)+\left(\mathrm{1}+\mathrm{1}+\mathrm{1}\right)+.... \\ $$$$\:\:\:\:{S}=\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:+\left(\mathrm{1}+\mathrm{1}\right)+..... \\ $$$$...... \\ $$$${subtracting} \\ $$$$\mathrm{0}=\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+.... \\ $$$$ \\ $$

Commented by Rio Michael last updated on 18/Aug/20

thank you so much sir.

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{sir}. \\ $$

Commented by Dwaipayan Shikari last updated on 18/Aug/20

But it is not the real one. It only works in Quantum world  It is divergent

$${But}\:{it}\:{is}\:{not}\:{the}\:{real}\:{one}.\:{It}\:{only}\:{works}\:{in}\:{Quantum}\:{world} \\ $$$${It}\:{is}\:{divergent} \\ $$

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