<s>[INST] Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$ Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$
My try:
we have $$\frac{13}{32}=\frac{2^2+3^2}{2^5}=\frac{1}{8}\left(1+(1.5)^2)\right)$$
Let $x=1.5$
Now consider the function $$f(x)=\frac{1+x^2}{8}-\ln x$$
$$f'(x)=\frac{x}{4}-\frac{1}{x}$$ So $f$ is Decreasing in $(0,2)$
any help here?
 [/INST] I think we can try with $$\log\frac{1+x}{1-x}=2\left(x+\frac{x^3}{3}+\dots\right)$$ and put $x=1/5$. The calculations are simple and one can estimate the error easily. Estimation of error gives $$\log\frac{1+x}{1-x}<2x+\frac{2x^3}{3}+\frac{2x^5}{5(1-x^2)}$$ With a little calculation of the easy variety (division by 2,3,5 etc) you can conclude that the right hand side of the above inequality for $x=1/5$ is less than $13/32$. 
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