无穷级数
ζ
(
s
)
=
∑
k
=
1
∞
1
k
s
{\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}}
无穷级数
极限比较审敛法 是判别级数 敛散性的一种方法。
假设存在两个级数
Σ
n
a
n
{\displaystyle \Sigma _{n}a_{n}}
Σ
n
b
n
{\displaystyle \Sigma _{n}b_{n}}
n
{\displaystyle n}
a
n
,
b
n
≥
0
{\displaystyle a_{n},b_{n}\geq 0}
如果
lim
n
→
∞
a
n
b
n
=
c
{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c}
0
<
c
<
∞
{\displaystyle 0<c<\infty }
对
lim
n
→
∞
a
n
b
n
=
c
{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c}
ε
>
0
{\displaystyle \varepsilon >0}
n
0
{\displaystyle n_{0}}
n
≥
n
0
{\displaystyle n\geq n_{0}}
|
a
n
b
n
−
c
|
<
ε
{\displaystyle \left|{\frac {a_{n}}{b_{n}}}-c\right|<\varepsilon }
−
ε
<
a
n
b
n
−
c
<
ε
{\displaystyle -\varepsilon <{\frac {a_{n}}{b_{n}}}-c<\varepsilon }
c
−
ε
<
a
n
b
n
<
c
+
ε
{\displaystyle c-\varepsilon <{\frac {a_{n}}{b_{n}}}<c+\varepsilon }
(
c
−
ε
)
b
n
<
a
n
<
(
c
+
ε
)
b
n
{\displaystyle (c-\varepsilon )b_{n}<a_{n}<(c+\varepsilon )b_{n}}
由于
c
>
0
{\displaystyle c>0}
ε
{\displaystyle \varepsilon }
c
−
ε
{\displaystyle c-\varepsilon }
b
n
<
1
c
−
ε
a
n
{\displaystyle b_{n}<{\frac {1}{c-\varepsilon }}a_{n}}
比较审敛法 ，如果
∑
n
a
n
{\displaystyle \sum _{n}a_{n}}
∑
n
b
n
{\displaystyle \sum _{n}b_{n}}
类似地，
a
n
<
(
c
+
ε
)
b
n
{\displaystyle a_{n}<(c+\varepsilon )b_{n}}
∑
n
b
n
{\displaystyle \sum _{n}b_{n}}
∑
n
a
n
{\displaystyle \sum _{n}a_{n}}
因此二者同时收敛或发散。
判断
∑
n
=
1
∞
1
n
2
+
2
n
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}+2n}}}
∑
n
=
1
∞
1
n
2
=
π
2
6
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}
由于
lim
n
→
∞
1
n
2
+
2
n
n
2
1
=
1
>
0
{\displaystyle \lim _{n\to \infty }{\frac {1}{n^{2}+2n}}{\frac {n^{2}}{1}}=1>0}
Rinaldo B. Schinazi: From Calculus to Analysis . Springer, 2011, ISBN 9780817682897 , pp. 50 （页面存档备份 ，存于互联网档案馆 ）
Michele Longo and Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series . Mathematics Magazine, Vol. 79, No. 3 (Jun., 2006), pp. 205–210 (JSTOR （页面存档备份 ，存于互联网档案馆 ）)
J. Marshall Ash: The Limit Comparison Test Needs Positivity . Mathematics Magazine, Vol. 85, No. 5 (December 2012), pp. 374–375 (JSTOR （页面存档备份 ，存于互联网档案馆 ）) 
