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Solutions to homework # 6.
1. If either lim sup on the righthand side is +
∞
, then the inequality is trivially satisﬁed.
Also, if lim sup
n
→∞
a
n
=
∞
, then (
a
n
) tends to
∞
; if lim sup
n
→∞
b
n
<
∞
, then the
sequence
a
n
+
b
n
tends to
∞
as well. So, it remains to consider the case when both
lim sup
a
n
and lim sup
b
n
are ﬁnite.
Take any subsequence (
a
n
k
+
b
n
k
) that tends to lim sup
n
→∞
(
a
n
+
b
n
). Since (
a
n
k
) is a
subsequence of (
a
n
), we conclude that lim sup
k
→∞
a
n
k
≤
lim sup
n
→∞
a
n
. Let (
a
n
k
l
) be the
subsequence of
a
n
k
that tends to lim sup
k
→∞
a
n
k
. Since lim sup
l
→∞
b
n
k
l
≤
lim sup
n
→∞
b
n
, we
obtain
lim sup(
a
n
+
b
n
) = lim
l
→∞
(
a
n
k
l
+
b
n
k
l
)
≤
lim sup
a
n
+ lim sup
b
n
.
2. (a) Since
a
n
= 1
/
(
√
n
+ 1+
√
n
)
>
1
/
(3
√
n
) and the series
∑
n
1
/
√
n
diverges, we conclude
by the comparison test that the series
∑
a
n
diverges as well.
(b) Since
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 Spring '09

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