Statement
edit
Proof
edit
Let
S
n
=
∑
k
=
1
n
a
k
b
k
{\textstyle S_{n}=\sum _{k=1}^{n}a_{k}b_{k}}
and
B
n
=
∑
k
=
1
n
b
k
{\textstyle B_{n}=\sum _{k=1}^{n}b_{k}}
.
From summation by parts , we have that
S
n
=
a
n
+
1
B
n
+
∑
k
=
1
n
B
k
(
a
k
−
a
k
+
1
)
{\textstyle S_{n}=a_{n+1}B_{n}+\sum _{k=1}^{n}B_{k}(a_{k}-a_{k+1})}
. Since
B
n
{\displaystyle B_{n}}
is bounded by M and
a
n
→
0
{\displaystyle a_{n}\to 0}
, the first of these terms approaches zero,
a
n
+
1
B
n
→
0
{\displaystyle a_{n+1}B_{n}\to 0}
as
n
→
∞
{\displaystyle n\to \infty }
.
We have, for each k ,
|
B
k
(
a
k
−
a
k
+
1
)
|
≤
M
|
a
k
−
a
k
+
1
|
{\displaystyle |B_{k}(a_{k}-a_{k+1})|\leq M|a_{k}-a_{k+1}|}
.
Since
(
a
n
)
{\displaystyle (a_{n})}
is monotone, it is either decreasing or increasing:
If
(
a
n
)
{\displaystyle (a_{n})}
is decreasing,
∑
k
=
1
n
M
|
a
k
−
a
k
+
1
|
=
∑
k
=
1
n
M
(
a
k
−
a
k
+
1
)
=
M
∑
k
=
1
n
(
a
k
−
a
k
+
1
)
,
{\displaystyle \sum _{k=1}^{n}M|a_{k}-a_{k+1}|=\sum _{k=1}^{n}M(a_{k}-a_{k+1})=M\sum _{k=1}^{n}(a_{k}-a_{k+1}),}
which is a telescoping sum that equals
M
(
a
1
−
a
n
+
1
)
{\displaystyle M(a_{1}-a_{n+1})}
and therefore approaches
M
a
1
{\displaystyle Ma_{1}}
as
n
→
∞
{\displaystyle n\to \infty }
. Thus,
∑
k
=
1
∞
M
(
a
k
−
a
k
+
1
)
{\textstyle \sum _{k=1}^{\infty }M(a_{k}-a_{k+1})}
converges.
If
(
a
n
)
{\displaystyle (a_{n})}
is increasing,
∑
k
=
1
n
M
|
a
k
−
a
k
+
1
|
=
−
∑
k
=
1
n
M
(
a
k
−
a
k
+
1
)
=
−
M
∑
k
=
1
n
(
a
k
−
a
k
+
1
)
,
{\displaystyle \sum _{k=1}^{n}M|a_{k}-a_{k+1}|=-\sum _{k=1}^{n}M(a_{k}-a_{k+1})=-M\sum _{k=1}^{n}(a_{k}-a_{k+1}),}
which is again a telescoping sum that equals
−
M
(
a
1
−
a
n
+
1
)
{\displaystyle -M(a_{1}-a_{n+1})}
and therefore approaches
−
M
a
1
{\displaystyle -Ma_{1}}
as
n
→
∞
{\displaystyle n\to \infty }
. Thus, again,
∑
k
=
1
∞
M
(
a
k
−
a
k
+
1
)
{\textstyle \sum _{k=1}^{\infty }M(a_{k}-a_{k+1})}
converges.
So, the series
∑
k
=
1
∞
B
k
(
a
k
−
a
k
+
1
)
{\textstyle \sum _{k=1}^{\infty }B_{k}(a_{k}-a_{k+1})}
converges, by the absolute convergence test. Hence
S
n
{\displaystyle S_{n}}
converges.
Applications
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Improper integrals
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Notes
edit
^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), pp. 253–255 Archived 2011-07-21 at the Wayback Machine .
^ "Where does the sum of $\sin(n)$ formula come from?".
References
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Hardy, G. H., A Course of Pure Mathematics , Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis , Marcel Dekker, Inc., New York, 1981. (§8.B.13–15) ISBN 0-8247-6949-X.
External links
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