## 2016年12月31日土曜日

### 数学 - 「離散的」な世界 - 数列 – 数列とその和 - その他の数列

1. $\begin{array}{l}\sum _{k=1}^{n}\frac{1}{\left(2k-1\right)\left(2k+1\right)}\\ \frac{1}{2}\sum _{i=1}^{n}\left(\frac{1}{2k-1}-\frac{1}{2k+1}\right)\\ =\frac{1}{2}\left(1-\frac{1}{2n+1}\right)\\ =\frac{n}{2n+1}\end{array}$

2. $\begin{array}{l}\sum _{k=1}^{n}\frac{1}{2}\left(\frac{1}{k}-\frac{1}{k+2}\right)\\ =\frac{1}{2}\left(\frac{1}{1}+\frac{1}{2}-\frac{1}{n-1+2}-\frac{1}{n+2}\right)\\ =\frac{1}{2}\left(\frac{3}{2}-\frac{1}{n+1}-\frac{1}{n+2}\right)\\ =\frac{1}{2}\left(\frac{3}{2}-\frac{2n+3}{\left(n+1\right)\left(n+2\right)}\right)\end{array}$

3. $\begin{array}{l}{a}_{k}=\frac{1}{\frac{k\left(k+1\right)}{2}}=\frac{2}{k\left(k+1\right)}=2\left(\frac{1}{k}-\frac{1}{k+1}\right)\\ 2\left(1-\frac{1}{n+1}\right)=\frac{2n}{n+1}\end{array}$

1. $\begin{array}{l}S=1+·2·2+3·{2}^{2}+4·{2}^{3}+5·{2}^{4}+···\\ 2S=0+2+2·{2}^{2}+3·{2}^{3}+4·{2}^{4}+···\\ -S=1+{2}^{1}+{2}^{2}+{2}^{3}+···+{2}^{n-1}-n·{2}^{n}\\ -S=\frac{1-{2}^{n}}{1-2}-n·{2}^{n}\\ S=-{2}^{n}+1+n·{2}^{n}\\ =\left(n-1\right)·{2}^{n}+1\end{array}$

2. $\begin{array}{l}S=1·2+3·{2}^{2}+5·{2}^{3}+···\\ 2S=0+1·{2}^{2}+3·{2}^{3}+···\\ -S=2+2·{2}^{2}+2·{2}^{3}+···+2·{2}^{n}-\left(2n-1\right)·{2}^{n+1}\\ S=-2-{2}^{3}·\frac{{2}^{n-1}-1}{2-1}+\left(2n-1\right){2}^{n+1}\\ =\left(2n-3\right){2}^{n+1}+6\end{array}$

3. $\begin{array}{l}S=\frac{1}{3}+\frac{3}{{3}^{2}}+\frac{5}{{3}^{3}}+···+\frac{2n-1}{{3}^{n}}\\ \frac{1}{3}S=0+\frac{1}{{3}^{2}}+\frac{3}{{3}^{3}}+···+\frac{2n-3}{{3}^{n}}+\frac{2n-1}{{3}^{n+1}}\\ \frac{2}{3}S=\frac{1}{3}+\frac{2}{{3}^{2}}+\frac{2}{{3}^{3}}+···++\frac{2}{{3}^{n}}-\frac{2n-1}{{3}^{n+1}}\\ \frac{2}{3}S=\frac{1}{3}-\frac{2n-1}{{3}^{n+1}}+\frac{2}{{3}^{2}}·\frac{1-\frac{1}{{3}^{n-1}}}{1-\frac{1}{3}}\\ S=\frac{1}{2}-\frac{2n-1}{2·{3}^{n}}+\frac{1}{3}·\frac{3}{2}·\frac{{3}^{n-1}-1}{{3}^{n-1}}\\ =\frac{1}{2}-\frac{2n-1}{2·{3}^{n}}+\frac{{3}^{n}-3}{2·{3}^{n}}\\ =\frac{1}{2}-\frac{2n-1}{2·{3}^{n}}+\frac{1}{2}-\frac{3}{2·{3}^{n}}\\ =1-\frac{n+1}{{3}^{n}}\end{array}$