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Lebl-Basic-Analysis-I
· Problem 2.6.2
Sequences and Series
Mathematics · cauchy-product
Problem
Suppose both
∑
n
=
0
∞
a
n
\sum_{n=0}^{\infty} a_n
∑
n
=
0
∞
a
n
and
∑
n
=
0
∞
b
n
\sum_{n=0}^{\infty} b_n
∑
n
=
0
∞
b
n
converge absolutely. Show that the product series,
∑
n
=
0
∞
c
n
\sum_{n=0}^{\infty} c_n
∑
n
=
0
∞
c
n
where
c
n
=
a
0
b
n
+
a
1
b
n
−
1
+
⋯
+
a
n
b
0
c_n = a_0 b_n + a_1 b_{n-1} + \cdots + a_n b_0
c
n
=
a
0
b
n
+
a
1
b
n
−
1
+
⋯
+
a
n
b
0
, also converges absolutely.
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