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VTRMC
2010 VTRMC
Problem 7
Problem 7
Part of
2010 VTRMC
Problems
(1)
convergence of sequence sum with 1/(na_n^2)
Source: VTRMC 2010 P7
5/17/2021
Let
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
be a convergent series of positive terms (so
a
i
>
0
a_i>0
a
i
>
0
for all
i
i
i
) and set
b
n
=
1
n
a
n
2
b_n=\frac1{na_n^2}
b
n
=
n
a
n
2
1
for
n
≥
1
n\ge1
n
≥
1
. Prove that
∑
n
=
1
∞
n
b
1
+
b
2
+
…
+
b
n
\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}
∑
n
=
1
∞
b
1
+
b
2
+
…
+
b
n
n
is convergent.
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