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votes
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How would you prove $\int^8_0\frac1{\sqrt{x+\frac1{\sqrt{x}}}}dx<4-\frac1{2019}$?
We would like to prove the following inequality.
$$\int^{8}_{0}\frac{1}{\sqrt{x+\frac{1}{\sqrt{x}}}}\,dx<4-\frac{1}{2019}\tag{1}$$
What I've tried is using the AM-GM inequality, $$x+\frac{1}{\...
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2
votes
1
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On the integral $\int_0^1 \frac{dx}{\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}}$ and the plastic constant
We have,
$$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
$$P=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\cdots}}}$$
with golden ratio $\phi$ and plastic constant $P$. If,
$$\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\...
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