# Is $$\log{2} = 0$$ ?

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

$\displaystyle \log(1 + x) = \sum_{i = 1}^{\infty}{\dfrac{(-1)^k x^k}{k}} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \dots \dots \\ \text{Putting x = 1} \\ \log{2} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots \dots \\ \log{2} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots - 2\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \dots \right) \\ \log{2} = \zeta(1) - \frac{2}{2}\left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots \right) \\ \log{2} = \zeta(1) - \zeta(1) \\ \log{2} = \boxed{0} \\ \textbf{Hence Proved}$

Note by Rajdeep Dhingra
3 years ago

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$$\zeta(1) = \infty$$

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

- 3 years ago

Yeah, absolutely right.

- 3 years ago

Thanks !

- 3 years ago

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

- 3 years ago

Thanks !

- 3 years ago

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

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

Staff - 3 years ago

Thanks a Lot

- 3 years ago

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

- 2 years, 11 months ago

- 3 years ago

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

- 3 years ago

I came up with this proof.

- 3 years ago

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Comment deleted Apr 14, 2015

Ok, delete the comments as they are all irrelevant and try all my new problems

- 3 years ago

On it.

- 3 years ago

Hey! Rajdeep are you in class 10 or 9.

- 3 years ago

Just came in Class 9

- 3 years ago

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

- 3 years ago

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

- 3 years ago

Thanks ! You can send snapshots.

- 3 years ago