crane boosted

${\displaystyle\sum_{k=1}^{\infty}}\dfrac{1}{k^2}=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+\dfrac{1}{8^2}+\dfrac{1}{9^2}+\dfrac{1}{10^2}+\dfrac{1}{11^2}+\dfrac{1}{12^2}+\dfrac{1}{13^2}+\dfrac{1}{14^2}+\dfrac{1}{15^2}+\dfrac{1}{16^2}+\dfrac{1}{17^2}+\dfrac{1}{18^2}+\dfrac{1}{19^2}+\dfrac{1}{20^2}+\dfrac{1}{21^2}+\dfrac{1}{22^2}+\dfrac{1}{23^2}+\dfrac{1}{24^2}+\dfrac{1}{25^2}+\dfrac{1}{26^2}+\dfrac{1}{27^2}+\dfrac{1}{28^2}+\cdots=\dfrac{\pi^2}{6}$

フィボナッチ数列の一般項
$a_n=\dfrac{1}{\sqrt{5}}\left\{(\dfrac{1+\sqrt{5}}{2})^n-(\dfrac{1-\sqrt{5}}{2})^n\right\}$

crane boosted

$a^{a^{a^{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}_{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}}_{a^{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}_{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}}}_{a^{a^{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}_{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}}_{a^{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}_{a^{a^{a^a_a}_{a^a_a}}_{a^{a^a_a}_{a^a_a}}}}}$

$\sum_{i=1}^n a_i\geq n\sqrt[n]{\prod_{i=1}^n a_i}$

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