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

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The discussion centers on proving that ln(2) is greater than (2/5)^(2/5). Participants are encouraged to engage with the problem as part of the Problem of the Week (POTW) initiative. The thread invites submissions and emphasizes community involvement in solving the mathematical challenge. A reminder to refer to the guidelines for submissions is included. Overall, the focus is on collaborative problem-solving within the math community.
anemone
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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!
 
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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)
 
Solution from other:

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

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