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Lebl-Basic-Analysis-I
· Problem 2.2.6
Sequences and Series
Mathematics · facts-about-limits-of-sequences
Problem
Let
x
n
:
=
1
n
2
x_n := \dfrac{1}{n^2}
x
n
:=
n
2
1
and
y
n
:
=
1
n
y_n := \dfrac{1}{n}
y
n
:=
n
1
. Define
z
n
:
=
x
n
y
n
z_n := \dfrac{x_n}{y_n}
z
n
:=
y
n
x
n
and
w
n
:
=
y
n
x
n
w_n := \dfrac{y_n}{x_n}
w
n
:=
x
n
y
n
. Do
{
z
n
}
n
=
1
∞
\{z_n\}_{n=1}^\infty
{
z
n
}
n
=
1
∞
and
{
w
n
}
n
=
1
∞
\{w_n\}_{n=1}^\infty
{
w
n
}
n
=
1
∞
converge? What are the limits? Can you apply Proposition 2.2.5? Why or why not?
Related theory
Sequences Convergence Cauchy Sequences Subsequences
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