If $lim_{ntoinfty}a_n=0$, then $lim_{ntoinfty}a_n^n=0$ [closed]
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Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.
I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.
sequences-and-series limits convergence infinity
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.
I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.
sequences-and-series limits convergence infinity
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
1
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02
add a comment |
Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.
I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.
sequences-and-series limits convergence infinity
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.
I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.
sequences-and-series limits convergence infinity
sequences-and-series limits convergence infinity
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
edited Dec 5 '18 at 17:03
gt6989b
33.3k22452
33.3k22452
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
asked Dec 5 '18 at 16:58
Mich2908Mich2908
24
24
closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
1
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02
add a comment |
1
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02
1
1
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02
add a comment |
2 Answers
2
active
oldest
votes
$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$
For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
add a comment |
HINT
Use the Root test on the sequence $b_n = a_n^n$.
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$
For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
add a comment |
$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$
For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
add a comment |
$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$
For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$
For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
answered Dec 5 '18 at 17:03
giannispapavgiannispapav
1,534324
1,534324
add a comment |
add a comment |
HINT
Use the Root test on the sequence $b_n = a_n^n$.
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
add a comment |
HINT
Use the Root test on the sequence $b_n = a_n^n$.
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
add a comment |
HINT
Use the Root test on the sequence $b_n = a_n^n$.
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
HINT
Use the Root test on the sequence $b_n = a_n^n$.
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
answered Dec 5 '18 at 17:02
gt6989bgt6989b
33.3k22452
33.3k22452
add a comment |
add a comment |
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1
What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02