【微分積分】1-1-1 数列の極限の定義｜問題集

\(\displaystyle |a_n-0|=\frac{1}{n^3}<\varepsilon\)
\(\displaystyle n^3>\frac{1}{\varepsilon}\)
\(\displaystyle n>\sqrt[3]{\frac{1}{\varepsilon}}\)
\(\displaystyle N(\varepsilon)=\left[\sqrt[3]{\frac{1}{\varepsilon}}\right]\)

\((a)\varepsilon=0.1\)のとき、
\(\displaystyle N(0.1)=\left[\sqrt[3]{\frac{1}{0.1}}\right]=[\sqrt[3]{10}]\approx[2.154]=3\)
\((b)\varepsilon=0.01\)のとき、
\(\displaystyle N(0.01)=\left[\sqrt[3]{\frac{1}{0.01}}\right]=[\sqrt[3]{100}]\approx[4.642]=5\)
\((c)\varepsilon=0.001\)のとき、
\(\displaystyle N(0.001)=\left[\sqrt[3]{\frac{1}{0.001}}\right]=[\sqrt[3]{1000}]=[10]=10\)

\(\displaystyle |b_n-0|=\frac{1}{2^n}<\varepsilon\)
\(\displaystyle 2^n>\frac{1}{\varepsilon}\)
\(\displaystyle n>\log_2\left(\frac{1}{\varepsilon}\right)\)
\(\displaystyle N(\varepsilon)=\left[\log_2\left(\frac{1}{\varepsilon}\right)\right]\)

\((a)\varepsilon=0.1\)のとき、
\(\displaystyle N(0.1)=\left[\log_2\left(\frac{1}{0.1}\right)\right]\)
\(=[\log_2{10}]\approx[3.32]\)
\(=4\)
\((b)\varepsilon=0.01\)のとき、
\(\displaystyle N(0.01)=\left[\log_2\left(\frac{1}{0.01}\right)\right]\)
\(=[\log_2{100}]\approx[6.64]\)
\(=7\)
\((c)\varepsilon=0.001\)のとき、
\(\displaystyle N(0.001)=\left[\log_2\left(\frac{1}{0.001}\right)\right]\)
\(=[\log_2{1000}]\approx[9.97]\)
\(=10\)

\(\displaystyle |c_n-0|=\frac{1}{n^2+1}<\varepsilon\)
\(\displaystyle n^2+1>\frac{1}{\varepsilon}\)
\(\displaystyle n>\sqrt{\frac{1}{\varepsilon}-1}\)
\(\displaystyle N(\varepsilon)=\left[\sqrt{\frac{1}{\varepsilon}-1}\right]\)

\((a)\varepsilon=0.1\)のとき、
\(\displaystyle N(0.1)=\left[\sqrt{\frac{1}{0.1}-1}\right]\)
\(=[\sqrt{10-1}]=[3]\)
\(=3\)
\((b)\varepsilon=0.01\)のとき、
\(\displaystyle N(0.01)=\left[\sqrt{\frac{1}{0.01}-1}\right]\)
\(=[\sqrt{100-1}]\approx[9.95]\)
\(=10\)
\((c)\varepsilon=0.001\)のとき、
\(\displaystyle N(0.001)=\left[\sqrt{\frac{1}{0.001}-1}\right]\)
\(=[\sqrt{1000-1}]\approx[31.6]\)
\(=32\)