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Question Number 104296 by Dwaipayan Shikari last updated on 20/Jul/20

$$$$$$$$$$\mathrm{FUN}\mathrm{TIME}\mathrm{AGAIN}!$$$$$$$$$$$$$$$${\mathrm{S}}_{\mathrm{n}}=1+2+3+4+5+6+7+8+9+..$$$${\mathrm{S}}_{\mathrm{n}}=1+(2+3+4)+(5+6+7)+...$$$${\mathrm{S}}_{\mathrm{n}}=1+9+18+27+...$$$${\mathrm{S}}_{\mathrm{n}}=1+9(1+2+3+4+5+6+7.......)$$$${\mathrm{S}}_{\mathrm{n}}=1+{9\mathrm{S}}_{\mathrm{n}}$$$${\mathrm{S}}_{\mathrm{n}}=-\frac{1}{8}$$$$$$$$$$$$$$$${\mathrm{S}}_{\mathrm{n}}=1-1+1-1+1-1+1-1+1-1+..$$$$\mathrm{S}=\frac{1}{2}$$$$$$$$$$$${\mathrm{S}}_{\mathrm{n}}=1-2+4-8+16-32+.....$$$${\mathrm{S}}_{\mathrm{n}}=\frac{1}{1+2}=\frac{1}{3}$$$$$$$$$$$${\mathrm{S}}_{\mathrm{n}}=1+2+4+8+16+...$$$${\mathrm{S}}_{\mathrm{n}}=1+2(1+2+4+8+...)$$$${\mathrm{S}}_{\mathrm{n}}=1+2(1+2(1+2+4+8+...)$$$${\mathrm{S}}_{\mathrm{n}}=1+2(1+{2\mathrm{S}}_{\mathrm{n}})$$$$-{3\mathrm{S}}_{\mathrm{n}}=3\Rightarrow {\mathrm{S}}_{\mathrm{n}}=-1$$

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