If $lim_{ntoinfty}a_n=0$, then $lim_{ntoinfty}a

If $lim_{ntoinfty}a_n=0$, then $lim_{ntoinfty}a_n^n=0$ [closed]

-2

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.

edited Dec 5 '18 at 17:03

gt6989b

33.3k22452

asked Dec 5 '18 at 16:58

Mich2908

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 |

-2

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.

edited Dec 5 '18 at 17:03

gt6989b

33.3k22452

asked Dec 5 '18 at 16:58

Mich2908

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 |

-2

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.

edited Dec 5 '18 at 17:03

gt6989b

33.3k22452

asked Dec 5 '18 at 16:58

Mich2908

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

Mich2908

edited Dec 5 '18 at 17:03

gt6989b

33.3k22452

asked Dec 5 '18 at 16:58

Mich2908

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

asked Dec 5 '18 at 16:58

Mich2908

asked Dec 5 '18 at 16:58

Mich2908

asked Dec 5 '18 at 16:58

Mich2908

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

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

giannispapav

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

gt6989b

33.3k22452

add a comment |

2 Answers
2

active

oldest

votes

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

giannispapav

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

giannispapav

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

giannispapav

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

giannispapav

1,534324

answered Dec 5 '18 at 17:03

giannispapav

1,534324

answered Dec 5 '18 at 17:03

giannispapav

1,534324

answered Dec 5 '18 at 17:03

giannispapav

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

gt6989b

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

gt6989b

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

gt6989b

33.3k22452

HINT

Use the Root test on the sequence $b_n = a_n^n$.

answered Dec 5 '18 at 17:02

gt6989b

33.3k22452

answered Dec 5 '18 at 17:02

gt6989b

33.3k22452

answered Dec 5 '18 at 17:02

gt6989b

33.3k22452

answered Dec 5 '18 at 17:02

gt6989b

33.3k22452

add a comment |

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