Is ln(2) Greater Than (2/5)^(2/5)?

[anemone]

Here is this week's POTW:

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Prove $\ln 2>\left(\dfrac{2}{5}\right)^{\frac{2}{5}}$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Hi MHB,

I will give the community another week to attempt at last week's high school POTW. I welcome anyone of you who are interested in this problem to give it one more try and I am looking forward to receiving your submission!(Happy)

 

[anemone]

Solution from other:

Spoiler

Summing just the $k=0$ and $k=1$ terms from the identity $$\ln 2=\sum_{k=0}^\infty \dfrac{2}{2k+1}\left(\dfrac{7}{31^{2k+1}}+\dfrac{3}{161^{2k+1}}+\dfrac{5}{49^{2k+1}}\right)$$ gives

$\ln 2>\dfrac{29558488681560}{42643891494953}\\ \ln2>0.693147\\ (\ln2)^5>(0.693147)^5 \\ (\ln 2)^5>0.160002 \\ (\ln 2)^5>\dfrac{4}{25}\\ \ln2>\left(\dfrac{2}{5}\right)^{\frac{2}{5}}$