Show that the following ten series are convergent:
1.
1
1
2
+
1
2
2
+
1
3
2
+
⋯
{\displaystyle \;{\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }
6.
1
+
1
2
2
+
1
3
3
+
1
4
4
+
⋯
{\displaystyle \;1+{\frac {1}{2{\sqrt {2}}}}+{\frac {1}{3{\sqrt {3}}}}+{\frac {1}{4{\sqrt {4}}}}+\cdots }
2.
1
2
+
2
2
2
+
3
2
3
+
4
2
4
+
⋯
{\displaystyle \;{\frac {1}{2}}+{\frac {2}{2^{2}}}+{\frac {3}{2^{3}}}+{\frac {4}{2^{4}}}+\cdots }
7.
1
−
1
3
2
+
1
5
2
−
1
7
2
+
1
9
2
−
⋯
{\displaystyle \;1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots }
3.
1
1
⋅
2
+
1
3
⋅
4
+
1
5
⋅
6
+
⋯
{\displaystyle \;{\frac {1}{1\cdot 2}}+{\frac {1}{3\cdot 4}}+{\frac {1}{5\cdot 6}}+\cdots }
8.
1
2
−
1
2
⋅
1
2
2
+
1
3
⋅
1
2
3
−
1
4
⋅
1
2
4
+
⋯
{\displaystyle \;{\frac {1}{2}}-{\frac {1}{2}}\cdot {\frac {1}{2^{2}}}+{\frac {1}{3}}\cdot {\frac {1}{2^{3}}}-{\frac {1}{4}}\cdot {\frac {1}{2^{4}}}+\cdots }
4.
1
3
+
1
⋅
3
3
⋅
6
+
1
⋅
3
⋅
5
3
⋅
6
⋅
9
+
⋯
{\displaystyle \;{\frac {1}{3}}+{\frac {1\cdot 3}{3\cdot 6}}+{\frac {1\cdot 3\cdot 5}{3\cdot 6\cdot 9}}+\cdots }
9.
1
log
2
−
1
log
3
+
1
log
4
−
⋯
{\displaystyle \;{\frac {1}{\log 2}}-{\frac {1}{\log 3}}+{\frac {1}{\log 4}}-\cdots }
5.
1
3
!
+
1
5
!
+
1
7
!
+
⋯
{\displaystyle \;{\frac {1}{3!}}+{\frac {1}{5!}}+{\frac {1}{7!}}+\cdots }
10.
1
2
2
+
1
3
2
+
1
4
2
+
⋯
{\displaystyle \;{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }
Show that the following four series are divergent :
11.
1
2
+
1
4
+
1
6
+
⋯
{\displaystyle \;{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{6}}+\cdots }
13.
2
!
10
+
3
!
10
2
+
4
!
10
3
+
⋯
{\displaystyle \;{\frac {2!}{10}}+{\frac {3!}{10^{2}}}+{\frac {4!}{10^{3}}}+\cdots }
12.
1
+
1
+
2
1
+
2
2
+
1
+
3
1
+
3
2
+
1
+
4
1
+
4
2
+
⋯
{\displaystyle \;1+{\frac {1+2}{1+2^{2}}}+{\frac {1+3}{1+3^{2}}}+{\frac {1+4}{1+4^{2}}}+\cdots }
14.
1
+
1
3
+
1
5
+
1
7
+
⋯
{\displaystyle \;1+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots }
142. Power series.
x
{\displaystyle x}
(A )
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
⋯
{\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots }
where the coefficients,
a
0
,
a
1
,
a
2
,
⋯
{\displaystyle a_{0},a_{1},a_{2},\cdots }
x
{\displaystyle x}
power series in x . Such series are of prime importance in the further study of the Calculus.
In special cases a power series in
x
{\displaystyle x}
x
{\displaystyle x}
x
{\displaystyle x}
x
{\displaystyle x}
(A ) only for the case when the coefficients are such that
lim
n
→
∞
(
a
n
+
1
a
n
)
=
L
{\displaystyle \lim _{n\to \infty }\left({\frac {a_{n+1}}{a_{n}}}\right)=L}
where
L
{\displaystyle L}
(A )
lim
n
→
∞
(
u
n
+
1
u
n
)
=
lim
n
→
∞
(
a
n
+
1
x
n
+
1
a
n
x
n
)
=
lim
n
→
∞
(
a
n
+
1
a
n
)
⋅
x
=
L
x
{\displaystyle \lim _{n\to \infty }\left({\frac {u_{n+1}}{u_{n}}}\right)=\lim _{n\to \infty }\left({\frac {a_{n+1}x^{n+1}}{a_{n}x^{n}}}\right)=\lim _{n\to \infty }\left({\frac {a_{n+1}}{a_{n}}}\right)\cdot x=Lx}
Referring to tests I , II , III , in §141 , we have in this case
p
=
L
x
{\displaystyle p=Lx}
(A ) is
I. Absolutely convergent when
|
L
x
|
<
1
{\displaystyle |Lx|<1}
|
x
|
<
|
1
L
|
{\displaystyle |x|<|{\tfrac {1}{L}}|}
II. Divergent when
|
L
x
|
>
1
{\displaystyle |Lx|>1}
|
x
|
>
|
1
L
|
{\displaystyle |x|>|{\tfrac {1}{L}}|}
III. No test when
|
L
x
|
=
1
{\displaystyle |Lx|=1}
x
=
|
1
L
|
{\displaystyle x=|{\tfrac {1}{L}}|}
