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Monotone Increasing Criteria
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Suppose I have a sequence $a_n$. To prove the sequence is monotone increasing, is it sufficient for me to prove that $a_n + 1/a_n > 1$ or $a_n + 1 - a_n > 0$ for arbitrary $n$? I believe the answer is yes; but, one example I am looking at uses induction. Is induction necessary?
real-analysis
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up vote
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Suppose I have a sequence $a_n$. To prove the sequence is monotone increasing, is it sufficient for me to prove that $a_n + 1/a_n > 1$ or $a_n + 1 - a_n > 0$ for arbitrary $n$? I believe the answer is yes; but, one example I am looking at uses induction. Is induction necessary?
real-analysis
asked 4 hours ago
Hat
841115
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Suppose I have a sequence $a_n$. To prove the sequence is monotone increasing, is it sufficient for me to prove that $a_n + 1/a_n > 1$ or $a_n + 1 - a_n > 0$ for arbitrary $n$? I believe the answer is yes; but, one example I am looking at uses induction. Is induction necessary?
real-analysis
asked 4 hours ago
Hat
841115
Suppose I have a sequence $a_n$. To prove the sequence is monotone increasing, is it sufficient for me to prove that $a_n + 1/a_n > 1$ or $a_n + 1 - a_n > 0$ for arbitrary $n$? I believe the answer is yes; but, one example I am looking at uses induction. Is induction necessary?
real-analysis
real-analysis
asked 4 hours ago
Hat
841115
asked 4 hours ago
Hat
841115
asked 4 hours ago
Hat
841115
asked 4 hours ago
Hat
841115
asked 4 hours ago
Hat
841115
841115
add a comment |Â
add a comment |Â
1 Answer
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If you show $a_n+1-a_n>0$ for all $n$ then clearly it is monotone increasing, nothing to prove here. As for $fraca_n+1a_n>1$ you have to be more careful. Look at the sequence $a_n=-n$. Then $fraca_n+1a_n=fracn+1n>1$ for all $n$ but the sequence is actually monotone decreasing. If you have a sequence of positive numbers though then yes, showing $fraca_n+1a_n>1$ for all $n$ is enough, no need to use induction.
answered 4 hours ago
Mark
3,434113
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
If you show $a_n+1-a_n>0$ for all $n$ then clearly it is monotone increasing, nothing to prove here. As for $fraca_n+1a_n>1$ you have to be more careful. Look at the sequence $a_n=-n$. Then $fraca_n+1a_n=fracn+1n>1$ for all $n$ but the sequence is actually monotone decreasing. If you have a sequence of positive numbers though then yes, showing $fraca_n+1a_n>1$ for all $n$ is enough, no need to use induction.
answered 4 hours ago
Mark
3,434113
add a comment |Â
up vote
4
down vote
accepted
If you show $a_n+1-a_n>0$ for all $n$ then clearly it is monotone increasing, nothing to prove here. As for $fraca_n+1a_n>1$ you have to be more careful. Look at the sequence $a_n=-n$. Then $fraca_n+1a_n=fracn+1n>1$ for all $n$ but the sequence is actually monotone decreasing. If you have a sequence of positive numbers though then yes, showing $fraca_n+1a_n>1$ for all $n$ is enough, no need to use induction.
answered 4 hours ago
Mark
3,434113
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
If you show $a_n+1-a_n>0$ for all $n$ then clearly it is monotone increasing, nothing to prove here. As for $fraca_n+1a_n>1$ you have to be more careful. Look at the sequence $a_n=-n$. Then $fraca_n+1a_n=fracn+1n>1$ for all $n$ but the sequence is actually monotone decreasing. If you have a sequence of positive numbers though then yes, showing $fraca_n+1a_n>1$ for all $n$ is enough, no need to use induction.
answered 4 hours ago
Mark
3,434113
If you show $a_n+1-a_n>0$ for all $n$ then clearly it is monotone increasing, nothing to prove here. As for $fraca_n+1a_n>1$ you have to be more careful. Look at the sequence $a_n=-n$. Then $fraca_n+1a_n=fracn+1n>1$ for all $n$ but the sequence is actually monotone decreasing. If you have a sequence of positive numbers though then yes, showing $fraca_n+1a_n>1$ for all $n$ is enough, no need to use induction.
answered 4 hours ago
Mark
3,434113
answered 4 hours ago
Mark
3,434113
answered 4 hours ago
Mark
3,434113
answered 4 hours ago
Mark
3,434113
3,434113
add a comment |Â
add a comment |Â
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