Is log2=0\log{2} = 0 ?

Here is a proof that log2\log{2} = 0. This is not true but can you spot the wrong step in it.


log(1+x)=i=1(1)kxkk=xx22+x33x44+x55x66+Putting x = 1log2=112+1314+1516+log2=1+12+13+14+15+2(12+14+16+18+110+)log2=ζ(1)22(1+12+13+14+15+)log2=ζ(1)ζ(1)log2=0Hence Proved\displaystyle \log(1 + x) = \sum_{i = 1}^{\infty}{\dfrac{(-1)^k x^k}{k}} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \dots \dots \\ \text{Putting x = 1} \\ \log{2} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots \dots \\ \log{2} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots - 2\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \dots \right) \\ \log{2} = \zeta(1) - \frac{2}{2}\left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots \right) \\ \log{2} = \zeta(1) - \zeta(1) \\ \log{2} = \boxed{0} \\ \textbf{Hence Proved}

Note by Rajdeep Dhingra
4 years, 4 months ago

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ζ(1)=\zeta(1) = \infty

ζ(1)ζ(1)=0\displaystyle \zeta(1) - \zeta(1) = \infty - \infty \neq 0

Krishna Sharma - 4 years, 4 months ago

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Thanks !

Rajdeep Dhingra - 4 years, 4 months ago

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Yeah, absolutely right.

Sandeep Bhardwaj - 4 years, 4 months ago

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ζ(1) \zeta(1) does not converge. So the statement ζ(1)ζ(1)=0 \zeta (1) - \zeta (1) =0 is like saying =0 \infty - \infty = 0 which is wrong because it is indeterminate.

Sudeep Salgia - 4 years, 4 months ago

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Thanks !

Rajdeep Dhingra - 4 years, 4 months ago

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Watch the limits in which the Taylor series converges. We are given that x<1 | x | < 1 , and hence we cannot apply this to the case where x=1 x = 1 or x=1 x = -1 , to find the value of log2 \log 2 or log0 \log 0 .

Similarly, we cannot apply the Geometric Progression sum of 11x=1+x+x2+x3+ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots to conclude that 10=1+1+1+1+ \frac{1}{0} = 1 + 1 + 1 + 1 + \ldots or that 12=11+11+ \frac{1}{2} = 1 - 1 + 1 - 1 + \ldots , because x=1,1 x = 1, -1 are out of the range in which the formula applies.

Calvin Lin Staff - 4 years, 4 months ago

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Thanks a Lot

Rajdeep Dhingra - 4 years, 4 months ago

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Rajdeep did u understand all this??? It was really out of my mind!!

sarvesh dubey - 4 years, 4 months ago

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I came up with this proof.

Rajdeep Dhingra - 4 years, 4 months ago

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@Rajdeep Dhingra Hey! Rajdeep are you in class 10 or 9.

Kalash Verma - 4 years, 4 months ago

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@Kalash Verma Just came in Class 9

Rajdeep Dhingra - 4 years, 4 months ago

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@Rajdeep Dhingra You are really a prodigy. How and when did you learn all this ?

Manish Dash - 4 years, 4 months ago

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@Rajdeep Dhingra Yep! I totally agree with @Manish Dash you are a total genius. In Open proof contest can I send snapshots or a word file.

Kalash Verma - 4 years, 4 months ago

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@Kalash Verma Thanks ! You can send snapshots.

Rajdeep Dhingra - 4 years, 4 months ago

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Alternatively, this works as a proof that the harmonic series diverges.

Akiva Weinberger - 4 years, 3 months ago

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