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Lebl-Basic-Analysis-I
· Problem 2.6.12
Sequences and Series
Mathematics · alternating-series
Problem
(Challenging) Show that the monotonicity in the alternating series test is necessary. That is, find a sequence of positive real numbers
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^{\infty}
{
x
n
}
n
=
1
∞
with
lim
n
→
∞
x
n
=
0
\lim_{n\to\infty} x_n = 0
lim
n
→
∞
x
n
=
0
but such that
∑
n
=
1
∞
(
−
1
)
n
x
n
\sum_{n=1}^{\infty} (-1)^n x_n
∑
n
=
1
∞
(
−
1
)
n
x
n
diverges.
Related theory
The Alternating Series Test
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