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1+2+3+4+5+........Infinite=? |
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Answer» Hey! We can surely say that the sum will be infinity. But if we go on fun experiment then we will get the sum as -1/12. How? Let's take sum be S S=1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …= ? S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … Take a negitive S1 -S1 = − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … Now 1 - S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … So , 1 - S1 = S1 Hence S1 = 1/2 S2 =1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + … S2 + S2 =1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + … + 0+ 1 − 2 + 3 − 4 + 5 − 6 + 7 + … =1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = S1 So, 2S2 = S1 2S2 = 1/2 S2 = 1/4 S − S2 =1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + … − 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 + … =0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + … = 4( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + … ) = 4S So , S - S2 = 4S Finally , S – 1/4 = 4S ⇒ S = – 1/12 S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …= – 1/12 Hence proved. But this answer is absolutely wrong as it breaks the sum law. |
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