vault backup: 2023-03-24 12:29:32

AiSD/Ćwiczenia/2. Ćwiczenia.md

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+$$\cases{T(0)=3\\ 3T(n)=nT(n-1)+2n!}
+$$
+|n|0|1|2|
+|-|-|-|-|
+|T(n)|$3$|$\frac{5}{3}$|$\frac{22}{9}$
+
+$$\begin{gather}
+
+a_n=3\\
+b_n=n\\
+c_n=2n!\\
+s_{n}=\cfrac{3^{n-1}}{n!}\\
+T_{n}=\cfrac{1}{\frac{3^{n-1}}{n!}\cdot 3} \cdot\left(1\cdot1\cdot3+\sum\limits_{k=1}^{n}\frac{3^{k-1}}{k!}\cdot 2k!\right)=\frac{n!}{3^{n}}\left(3+\sum\limits_{k=1}^{n}3^{k-1}\cdot \frac{2}{3}\right)=\\=\frac{n!}{3^{n}}\left(3+ \frac{2}{3}\cdot \frac{3-3^{n-1}}{1-3}\right)=\frac{n!}{3^{n}}\left(2+3^{n}\right)
+\end{gather}$$