We want to use the Alternating Series Test to determine if the series:
∑
k
=
1
∞
(
−
1
)
2
k
+
1
k
5
k
11
+
9
\displaystyle {\sum_{{{k}={1}}}^{\infty}}\ {\left(-{1}\right)}^{{{2}{k}+{1}}}\frac{{{k}^{{{5}}}}}{{\sqrt{{{k}^{{{11}}}+{9}}}}}
k
=
1
∑
∞
(
−
1
)
2
k
+
1
k
11
+
9
k
5
converges or diverges.
We can conclude that:
The Alternating Series Test does not apply because the absolute value of the terms do not approach 0, and the series diverges for the same reason.
The Alternating Series Test does not apply because the terms of the series do not alternate.
The Alternating Series Test does not apply because the absolute value of the terms are not decreasing.
The series converges by the Alternating Series Test.
The series diverges by the Alternating Series Test.
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