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Question Number 120711 by TITA last updated on 02/Nov/20

$$showthat1+2+3+4...=\frac{-1}{8}$$

Answered by Dwaipayan Shikari last updated on 02/Nov/20

$$S=1+2+3+4+5+6+...$$$$S=1+(2+3+4)+(5+6+7)+(8+9+10)+...$$$$S=1+9+18+27+36+..$$$$S=1+9S$$$$-8S=1$$$$S=-\frac{1}{8}$$$$$$$$Or$$$$S=1+2+3+4+5+6+7+8+...$$$$-4S=-4-8-12-....$$$$-3S=1-2+3-4+6-8+....$$$$-3S=\frac{1}{4}$$$$S=-\frac{1}{12}$$$$$$$$Or\zeta (-1)=1+2+3+4+5+6+7+8+.....$$$$\zeta (1-2)=2^{1-2}{\pi }^{-2}cos\left(\frac{2\pi }{2}\right)\mathrm{\Gamma }\left(2\right)\zeta \left(2\right)$$$$\zeta (-1)=-\frac{1}{2{\pi }^2}.\frac{{\pi }^2}{6}=-\frac{1}{12}$$

Commented by Dwaipayan Shikari last updated on 02/Nov/20

$$Generally\zeta (1-s)=2^{1-s}{\pi }^{-s}cos\left(\frac{\pi s}{2}\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right)$$

Answered by Rohit143Jo last updated on 02/Nov/20

$$ans:-Let,1+2+3+4+5+6+7+8+9+10+............=S$$$$\Rightarrow 1+9+18+27+36...........=S[\because 2+3+4=9;5+6+7=18.....]$$$$\Rightarrow 1+9(1+2+3+4+.........)=S$$$$\Rightarrow 1+9S=S$$$$\Rightarrow -8S=1$$$$\Rightarrow S=(-1/8)$$

Commented by TITA last updated on 02/Nov/20

$$thanks$$

Answered by Rohit143Jo last updated on 02/Nov/20

$$ans:-Let,1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+.........=S$$$$\Rightarrow 1+2+25+50+75+........=S[\because 3+4+5+6+7=25;8+9+10+11+12=50...........]$$$$\Rightarrow 3+25(1+2+3+4+......)=S$$$$\Rightarrow 3+25S=S$$$$\Rightarrow -24S=3$$$$\Rightarrow S=(-3/24)$$$$\Rightarrow S=(-1/8)$$

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