# 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
5 years, 2 months ago

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$\zeta(1) = \infty$

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

- 5 years, 2 months ago

Thanks !

- 5 years, 2 months ago

Yeah, absolutely right.

- 5 years, 2 months 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.

- 5 years, 2 months ago

Thanks !

- 5 years, 2 months 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 - 5 years, 2 months ago

Thanks a Lot

- 5 years, 2 months ago

- 5 years, 2 months ago

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

- 5 years, 2 months ago

I came up with this proof.

- 5 years, 2 months ago

Hey! Rajdeep are you in class 10 or 9.

- 5 years, 2 months ago

Just came in Class 9

- 5 years, 2 months ago

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

- 5 years, 2 months 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.

- 5 years, 2 months ago

Thanks ! You can send snapshots.

- 5 years, 2 months ago

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

- 5 years, 1 month ago