Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

This problem seems to be so hard to prove "elegantly" without aid of calculator, is because you have managed to find such a good approximation! One would need to estimate really well to be able to prove that one is greater than the other.

Continued fractions can be used to give a proof:

$$\left(\dfrac{2}{5}\right)^2 = \dfrac{4}{25} = \cfrac{1}{6 + \cfrac{1}{4}}$$ is a convergent of the the continued fraction of $$(\log 2)^5 = [0; 6, 4, 592, 1, \dots]$$ If you take the CF of $(\log 2)^5$ for granted, the proof of what you want falls right out: $\frac{4}{25}$ is a convergent which is smaller (the convergents alternate bigger/smaller).

Note the apperance of the huge $592$ term, which tells you that $\dfrac{4}{25}$ will be a good approximation, because the convergent corresponding to the $592$ term is greater than $(\log 2)^5$.

Perhaps a more compelling "reason" for it being a good approximation is that the continued fractions of $\log 2 = [0; 1, 2, 3, 1, 6, 3, 1, \dots]$ and $\left(\frac{2}{5}\right)^{2/5} = [0; 1, 2, 3, 1, 6, 3, 2,\dots]$, match up to a good $6$ terms! The $7^{\text{th}}$ term tells you that $\log 2$ must be greater: the parity of the position where two CFs first differ determines if the one with the greater number in that position is greater or not (which also explains the alternating property mentioned above).

If you absolutely want a proof which needs no calculator (i.e. can be verified manually in an hour or so :-)), here is one (with calculations missing):

Take the power series

$$\log (1+x) - \log (1-x) = 2\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}$$

and set $x = \frac{1}{3}$, you get $\log 2$ on the left.

Now you can truncate the power series at any point, and get a smaller number than $\log (1+x) - \log (1-x)$ , as coefficients are all positive.

Now if you truncate the series at $n=4$ (include $n=4$ term) you get the value $\dfrac{4297606}{6200145}$ which is greater than $\left(\dfrac{2}{5}\right)^{2/5}$. This you can see (manually) by computing the fifth power and computing the (numerator of the) difference of the two fractions you get.

I won't go into more details, as they are quite tedious to do so completely manually without the aid of any calculators, and this is probably not what you were expecting anyway.

Interestingly, it might be easier to do the computations if you work in base $9$ or $3$ (because of the powers of $\frac{1}{3}$, you can quickly read off some of the digits, like spigot algorithms).

$$ \left(2 \over 5\right)^{2/5} = 0.69314\color{#ff0000}{\Large 4}843155146\ldots\,, \qquad\qquad \ln\left(2\right) = 0.69314\color{#ff0000}{\Large 7}180559945\ldots $$

So$\ldots$

Hmm, first I thought the following would allow to compute the proof mentally, but, well... although I find it a remarkable simplification I'll need the pocket calculator in the end. But let's see:

$$\ln(2) \gt (2 / 5)^{2 /5} = \left({16 \over 100 }\right)^{1/5} $$ We have also $$ \ln(2) = \ln \left( 1+1/3 \over 1-1/3 \right) = 2\left( {1\over3} + {1\over 3^3 \cdot 3} + {1 \over 3^5 \cdot 5}+ \cdots \right) \\ = {2 \over 3} \left( 1 + { 1 \over 9\cdot3 } + {1 \over 81 \cdot 5}+ \cdots ) \right) $$ Turn the factor 2/3 to the the rhs then $$ 1 + 1/27 + 1/81/5+ 1/729/7 \cdots \gt \left( {3^5 \over2 \cdot 100 }\right) ^{1/5}= \left( 1+{43\over 200}\right)^{1/5} $$

Now in general we have for a fifth root $$ (1+x)^{1/5} = 1 + x/5 - 2(x/5)^2 + 6(x/5)^3 - 21(x/5)^4 + 399/5 (x/5)^5 - \cdots $$

Thus we must evaluate $$ 1 + 1/27 + 1/81/5+ 1/729/7 \cdots \gt 1 +43/1000 - 2(43/1000)^2 + \cdots $$

"In principle" this can be done with paper & pen only because the terms decrease quickly, and some adaptions of denominators are possible, however, that was too tidy for me. I found using a calculator (Pari/GP) that we must evaluate the lhs with 4 terms and the rhs with 5 terms (of course excluding the 1's) to get the decision - because after that the partial sums in the lhs still increase but in the rhs decrease.