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Mathematics
Mathematics
Problem
For which of the following series will every possible rearrangement of its terms converge to the same sum as the original series? a)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
2
/
3
\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^{2/3}}
n
=
1
∑
∞
n
2/3
(
−
1
)
n
+
1
b)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
3
\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^{3}}
n
=
1
∑
∞
n
3
(
−
1
)
n
+
1
c)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n}
n
=
1
∑
∞
n
(
−
1
)
n
+
1
d)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
n
+
1
\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\dfrac{n}{n+1}
n
=
1
∑
∞
(
−
1
)
n
+
1
n
+
1
n
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