![$\lim_{n\to\infty}\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}=1$](/mathtex/64/64b5ee15218a745d47ad2368b9781ed5.gif "kopírovat do textarea")
, je puvodni limita typu 
a pro jeji vypocet pouzijes vzorec 
v aproximaci ![$\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2} \approx 1+\frac{a}{n}$](/mathtex/dc/dcba8792a0e89ca803ff9d9a3e09638c.gif "kopírovat do textarea")
, tzn. spocitat limitu![$a=\lim_{n\to\infty}\left(\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}-1\right)n = \lim_{n\to\infty}\frac{1}{2}\left(\frac{2^{\frac{1}{n}}-1}{\frac{1}{n}}+\frac{3^{\frac{1}{n}}-1}{\frac{1}{n}}\right) = \frac{1}{2}(\ln2+\ln3)=\ln\sqrt{6}$](/mathtex/59/59560dd3a52189153ec0794d3d6d1da5.gif "kopírovat do textarea")
![$\lim_{n\to\infty}\left(\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}\right)^n=\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=\cdots$](/mathtex/9f/9f88061b662d8834af91348793812568.gif "kopírovat do textarea")
#4 22. 10. 2018 20:42 — Editoval laszky (22. 10. 2018 21:22)
Re: Limity
↑ krakonoš:
Ahoj, neshodujem. Vysledek musi byt urcite vetsi, je to videt napr. uz z AG nerovnosti![$\left(\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}\right)^n \geq \left(\sqrt{\sqrt[n]{2}\sqrt[n]{3}}\right)^n = \sqrt{6} $](/mathtex/fb/fbe06d4630046b73bdc9536f269ab77c.gif "kopírovat do textarea")
![$\left(\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}\right)^n =\frac{\sum_{k=0}^n\binom{n}{k}2^{\frac{k}{n}}3^{\frac{n-k}{n}}}{2^n} \stackrel{YN}{\leq} \frac{\sum_{k=0}^n\binom{n}{k}\left(\frac{\left(2^{\frac{k}{n}}\right)^{\frac{n}{k}}}{\frac{n}{k}}+\frac{\left(3^{\frac{n-k}{n}}\right)^{\frac{n}{n-k}}}{\frac{n}{n-k}}\right)}{2^n} = $](/mathtex/5a/5aa5a99bc1b558d4e79c77305a1a9768.gif "kopírovat do textarea")
Offline
#5 22. 10. 2018 21:12
Re: Limity
↑ laszky[/re[re]p573633:Vysla mi odmocnina z sesti. Prve jsem prehledla omylem mocninu n v tom dlouhem retezci.
tg(x) je funkcí života.Jednou jsi nahoře🗽, podruhé zas dole 🗿.
Offline