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Mathematics
Problem
Consider the series
∑
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
. Which of the following two requirements fails for this series: (i)
{
b
n
}
\{b_n\}
{
b
n
}
is eventually non-increasing, or (ii)
lim
n
→
∞
b
n
=
0
\displaystyle \lim_{n\to\infty} b_n = 0
n
→
∞
lim
b
n
=
0
? Briefly justify your choice and state whether the series converges or diverges.
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