# 等諧數列

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## 性質

${\displaystyle a_{n}={\frac {1}{{\frac {1}{a}}+{\frac {n-1}{h}}}}}$

${\displaystyle \{a\,,\,\,{\frac {1}{{\frac {1}{a}}+{\frac {1}{h}}}}\,,\,\,{\frac {1}{{\frac {1}{a}}+{\frac {2}{h}}}}\,,\,\cdots \,,\,\,{\frac {1}{{\frac {1}{a}}+{\frac {n-1}{h}}}}\}}$

${\displaystyle h={\frac {1}{{\frac {1}{a_{n+1}}}-{\frac {1}{a_{n}}}}}}$

${\displaystyle h={\frac {m-n}{{\frac {1}{a_{m}}}-{\frac {1}{a_{n}}}}}}$

${\displaystyle {\frac {1}{a_{n-1}}}+{\frac {1}{a_{n+1}}}={\frac {2}{a_{n}}}}$

{\displaystyle {\begin{aligned}{\frac {1}{a_{n-1}}}+{\frac {1}{a_{n+1}}}&=\left({\frac {1}{a}}+{\frac {n-2}{h}}\right)+\left({\frac {1}{a}}+{\frac {n}{h}}\right)\\&={\frac {2}{a}}+{\frac {2n-2}{h}}\\&=2\left({\frac {1}{a}}+{\frac {n-1}{h}}\right)\\&={\frac {2}{a_{n}}}\\\end{aligned}}}

${\displaystyle a_{n}={\frac {2}{{\frac {1}{a_{n-1}}}+{\frac {1}{a_{n+1}}}}}}$

${\displaystyle {\frac {1}{a_{m}}}+{\frac {1}{a_{n}}}={\frac {1}{a_{p}}}+{\frac {1}{a_{q}}}}$

{\displaystyle {\begin{aligned}{\frac {1}{a_{m}}}+{\frac {1}{a_{n}}}&=\left({\frac {1}{a}}+{\frac {m-1}{h}}\right)+\left({\frac {1}{a}}+{\frac {n-1}{h}}\right)\\&={\frac {2}{a}}+{\frac {m+n-2}{h}}\\&={\frac {2}{a}}+{\frac {p+q-2}{h}}\\&=\left({\frac {1}{a}}+{\frac {p-1}{h}}\right)+\left({\frac {1}{a}}+{\frac {q-1}{h}}\right)\\&={\frac {1}{a_{p}}}+{\frac {1}{a_{q}}}\\\end{aligned}}}

${\displaystyle {\frac {1}{a_{n-k}}}+{\frac {1}{a_{n+k}}}={\frac {2}{a_{n}}}}$
${\displaystyle a_{n}={\frac {2}{{\frac {1}{a_{n-k}}}+{\frac {1}{a_{n+k}}}}}}$

• ${\displaystyle \{b\cdot a_{n}\}}$  是一個等諧數列。
• ${\displaystyle \{{\frac {b}{a_{n}}}\}}$  是一個等差數列

## 等諧数列和

${\displaystyle S_{n}=a\int _{0}^{1}{\left({\frac {1-x^{{\frac {a}{h}}\cdot n}}{1-x^{\frac {a}{h}}}}\right)}\mathrm {d} x}$

{\displaystyle {\begin{aligned}S_{n}&=a+{\frac {1}{{\frac {1}{a}}+{\frac {1}{h}}}}+{\frac {1}{{\frac {1}{a}}+{\frac {2}{h}}}}+\cdots +{\frac {1}{{\frac {1}{a}}+{\frac {n-1}{h}}}}\\&=a+{\frac {a}{1+{\frac {a}{h}}}}+{\frac {a}{1+{\frac {2a}{h}}}}+\cdots +{\frac {a}{1+{\frac {(n-1)a}{h}}}}\\&=a\left(1+{\frac {1}{{\frac {a}{h}}+1}}+{\frac {1}{{\frac {2a}{h}}+1}}+\cdots +{\frac {1}{{\frac {(n-1)a}{h}}+1}}\right)\\&=a\left[x+{\frac {x^{{\frac {a}{h}}+1}}{{\frac {a}{h}}+1}}+{\frac {x^{{\frac {2a}{h}}+1}}{{\frac {2a}{h}}+1}}+\cdots +{\frac {x^{{\frac {(n-1)a}{h}}+1}}{{\frac {(n-1)a}{h}}+1}}\right]_{x=0}^{x=1}\\&=a\int _{0}^{1}\left(1+x^{\frac {a}{h}}+x^{\frac {2a}{h}}+\cdots +x^{\frac {(n-1)a}{h}}\right)\mathrm {d} x\\&=a\int _{0}^{1}\left({\frac {1-x^{{\frac {a}{h}}\cdot n}}{1-x^{\frac {a}{h}}}}\right)\mathrm {d} x\\\end{aligned}}}

{\displaystyle {\begin{aligned}S_{4}&={\frac {1}{3}}\int _{0}^{1}\left({\frac {1-x^{{\frac {2}{3}}\cdot 4}}{1-x^{\frac {2}{3}}}}\right)\mathrm {d} x\\&={\frac {1}{3}}\int _{0}^{1}\left({\frac {1-x^{\frac {8}{3}}}{1-x^{\frac {2}{3}}}}\right)\mathrm {d} x\\&\approx 0.7873\end{aligned}}}

## 等諧数列积

${\displaystyle P_{n}=h^{n}\cdot {\frac {\Gamma ({\frac {h}{a}})}{\Gamma ({\frac {h}{a}}+n)}}}$

{\displaystyle {\begin{aligned}P_{n}&=a\cdot {\frac {1}{{\frac {1}{a}}+{\frac {1}{h}}}}\cdot {\frac {1}{{\frac {1}{a}}+{\frac {2}{h}}}}\cdot \cdots \cdot {\frac {1}{{\frac {1}{a}}+{\frac {n-1}{h}}}}\\&={\frac {1}{{\frac {1}{h^{n}}}\cdot {\frac {\Gamma ({\frac {h}{a}}+n)}{\Gamma ({\frac {h}{a}})}}}}\\&=h^{n}\cdot {\frac {\Gamma ({\frac {h}{a}})}{\Gamma ({\frac {h}{a}}+n)}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}P_{4}&={\frac {1}{2^{4}}}\cdot {\frac {\Gamma ({\frac {3}{2}})}{\Gamma ({\frac {3}{2}}+4)}}\\&={\frac {1}{16}}\cdot {\frac {0.88622\dots }{52.342\dots }}\\&={\frac {1}{945}}\\\end{aligned}}}