O limita cu sirul armonic
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O limita cu sirul armonic
Sa se calculeze \( \lim_{n\to\infty}(\sqrt[n] {n})^{H_{n}} \), unde \( H_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n} \).
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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Radu Titiu
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\( L=\lim_{n} \left( 1+\sqrt[n]{n}-1 \right)^{\frac{1}{\sqrt[n]{n}-1}\cdot (\sqrt[n]{n}-1)H_{n}}=e^{\lim_{n} (\sqrt[n]{n}-1)H_{n}} \)
\( \lim_{n} (\sqrt[n]{n}-1)H_{n}= \lim_{n}\frac{e^{\frac{ln(n)}{n}}-1}{\frac{ln(n)}{n}}\cdot \frac{ln(n)^2}{n}\frac{H_{n}}{ln(n)}=\lim_{n}\frac{ln(n)^2}{n}=0 \)
deci L=1
\( \lim_{n} (\sqrt[n]{n}-1)H_{n}= \lim_{n}\frac{e^{\frac{ln(n)}{n}}-1}{\frac{ln(n)}{n}}\cdot \frac{ln(n)^2}{n}\frac{H_{n}}{ln(n)}=\lim_{n}\frac{ln(n)^2}{n}=0 \)
deci L=1
A mathematician is a machine for turning coffee into theorems.