Analysis Ratio Test
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Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.
This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!
real-analysis analysis convergence
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up vote
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Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.
This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!
real-analysis analysis convergence
A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11
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up vote
2
down vote
favorite
up vote
2
down vote
favorite
Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.
This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!
real-analysis analysis convergence
Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.
This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!
real-analysis analysis convergence
edited Aug 11 at 1:17
zzuussee
1,701420
1,701420
asked Aug 11 at 0:55
Brad Scott
885
885
A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11
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A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11
A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11
A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11
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1 Answer
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Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus
$$fraca_n+1a_n<1$$
as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus
$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus
$$fraca_n+1a_n<1$$
as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus
$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.
add a comment |Â
up vote
2
down vote
accepted
Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus
$$fraca_n+1a_n<1$$
as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus
$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus
$$fraca_n+1a_n<1$$
as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus
$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.
Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus
$$fraca_n+1a_n<1$$
as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.
EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus
$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$
Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.
edited Aug 11 at 1:10
answered Aug 11 at 1:05
zzuussee
1,701420
1,701420
add a comment |Â
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A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/â¦
â zzuussee
Aug 11 at 1:11