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Lebl-Basic-Analysis-I
· Problem 2.5.7
Sequences and Series
Mathematics · series-convergence
Problem
(Challenging) Suppose
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^{\infty}
{
x
n
}
n
=
1
∞
is a decreasing sequence and
∑
n
=
1
∞
x
n
\sum_{n=1}^{\infty} x_n
∑
n
=
1
∞
x
n
converges. Prove
lim
n
→
∞
n
x
n
=
0
\lim_{n\to\infty} n x_n = 0
lim
n
→
∞
n
x
n
=
0
.
Related theory
Series Convergence Tests And Absolute Convergence
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