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Mathematics
Mathematics
Problem
Which of the following infinite series diverges? a)
∑
k
=
1
∞
(
−
1
)
k
1
k
\displaystyle \sum_{k=1}^{\infty} (-1)^{k}\,\frac{1}{k}
k
=
1
∑
∞
(
−
1
)
k
k
1
b)
∑
k
=
0
∞
(
4
3
)
k
\displaystyle \sum_{k=0}^{\infty} \left( \tfrac43 \right)^{k}
k
=
0
∑
∞
(
3
4
)
k
c)
∑
k
=
1
∞
1
k
2
\displaystyle \sum_{k=1}^{\infty} \frac{1}{k^{2}}
k
=
1
∑
∞
k
2
1
d)
∑
k
=
1
∞
k
2
k
\displaystyle \sum_{k=1}^{\infty} \frac{k}{2^{k}}
k
=
1
∑
∞
2
k
k
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