# Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ (https://math.stackexchange.com/a/1618454/134791)

prove that $$3<\pi<\frac{22}{7}$$

Is there a similar argument for the following $\log (2)$ inequality? $$\frac{2}{3}<\log(2)<\frac{7}{10}$$

• I was just searching my books for a proof that the value of pi we learn in school is slightly greater than pi. So, I'm glad I found this question ! Feb 6, 2016 at 6:38
• @user230452 You may find that proof here: en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80 Feb 6, 2016 at 6:56

There are positive integrals that relate $\log(2)$ to its first four convergents: $0,1,\frac{2}{3},\frac{7}{10}$. \begin{align} \int_0^1\frac{2x}{1+x^2}dx &= \log\left(2\right) \\ \int_0^1\frac{(1-x)^2}{1+x^2}dx &= 1-\log\left(2\right) \\ \int_0^1\frac{x^2(1-x)^2}{1+x^2}dx &= \log\left(2\right)-\frac{2}{3} \\ \int_0^1\frac{x^4(1-x)^2}{1+x^2}dx &=\frac{7}{10}-\log\left(2\right) \\ \end{align}

Therefore, $$-\int_0^1\frac{x^2(1-x)^2}{1+x^2}dx<0<\int_0^1\frac{x^4(1-x)^2}{1+x^2}dx$$

$$\frac{2}{3}-\log(2)<0<\frac{7}{10}-\log\left(2\right)$$

$$\frac{2}{3}<\log(2)<\frac{7}{10}$$

A similar set is available with denominators $(1+x)$:

\begin{align} \int_0^1 \frac{1}{1+x}dx &= \log(2) \\ \int_0^1 \frac{x}{1+x}dx &= 1-\log(2)\\ \frac{1}{2}\int_0^1 \frac{x^2(1-x)}{1+x} dx &= \log(2)-\frac{2}{3} \\ \frac{1}{2}\int_0^1 \frac{x^5(1-x)}{1+x} dx &= \frac{7}{10}-\log(2) \end{align}

and series versions are given by

\begin{align} \log(2)-\frac{2}{3} &= \sum_{k=1}^\infty \frac{1}{(2k+1)(2k+2)(2k+3)} \\ \frac{7}{10}-\log(2) &= \sum_{k=2}^\infty \frac{1}{(2k+2)(2k+3)(2k+4)} \\ \end{align}

• You answered your own question ? Feb 6, 2016 at 6:40
• You posted an answer at nearly the exact same time as you posted the question? Feb 6, 2016 at 7:40
• @qwr That's the "somewhere" I didn't manage to find when answering to user230452... thank you! Feb 6, 2016 at 7:47
• @user230452 The exponents in wolframalpha.com/input/… may be changed to find rational approximations to $log(2)$ and $\pi$. Feb 6, 2016 at 9:50