Sbírka řešených úloh

Limita geometrické posloupnosti II

Úloha číslo: 842

• Řešení

  Určujeme limitu posloupnosti

  \[\lim_{\small n\to\infty} \ \frac{1^n + 2^n + 3^n + 4^n + 5^n}{5{,}001^{n+1}}.\]

  Podle věty o aritmetice limit dostaneme

  \[\lim_{\small n\to\infty} \ \frac{1^n + 2^n + 3^n + 4^n + 5^n}{5{,}001^{n+1}} = \] \[ = \lim_{\small n\to\infty} \left(\frac{1^n}{5{,}001^{n+1}} + \frac{2^n}{5{,}001^{n+1}} + \frac{3^n}{5{,}001^{n+1}} + \frac{4^n}{5{,}001^{n+1}} + \frac{5^n}{5{,}001^{n+1}}\right) = \] \[ = \lim_{\small n\to\infty} \left(\frac{1^n}{5{,}001^{n+1}}\right) + \lim_{\small n\to\infty} \left(\frac{2^n}{5{,}001^{n+1}}\right) + \lim_{\small n\to\infty} \left(\frac{3^n}{5{,}001^{n+1}}\right) + \] \[ + \lim_{\small n\to\infty} \left(\frac{4^n}{5{,}001^{n+1}}\right) + \lim_{\small n\to\infty} \left(\frac{5^n}{5{,}001^{n+1}}\right) = \] \[ = \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{1^n}{5{,}001^{n}}\right) + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{2^n}{5{,}001^{n}}\right) + \frac{1}{5{,}001}\cdot\] \[\cdot\lim_{\small n\to\infty} \left(\frac{3^n}{5{,}001^{n}}\right) + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{4^n}{5{,}001^{n}}\right) + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{5^n}{5{,}001^{n}}\right) = \] \[ = \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{1}{5{,}001}\right)^n + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{2}{5{,}001}\right)^n + \frac{1}{5{,}001}\cdot\] \[\cdot\lim_{\small n\to\infty} \left(\frac{3}{5{,}001}\right)^n + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{4}{5{,}001}\right)^n + \frac{1}{5{,}001}\cdot\lim_{\small n\to\infty} \left(\frac{5}{5{,}001}\right)^n = \]

  a podle části (c) úlohy Limita geometrické posloupnosti dostaneme

  \[ = \frac{1}{5{,}001}\cdot 0 +\frac{1}{5{,}001}\cdot 0+\frac{1}{5{,}001}\cdot 0+\frac{1}{5{,}001}\cdot 0+\frac{1}{5{,}001}\cdot 0 = 0.\]