Proving convergence of a sequence with logarithm
I was wondering about the convergence of a sequence based on the convergence of the logarithm of the sequence. I can't seem to find anything online, and it seems to make intuitive sense, but I'm not entirely sure and am finding it difficult to show it.
To state the issue, if we have $log(a_n)$ converges, can it be shown that $a_n$ likewise converges to the value of $e$ raised to the limit?
Again, this seems somewhat intuitive, but I can't seem to figure out how to prove it effectively or find anything online. Any help would be appreciated.
real-analysis sequences-and-series logarithms
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
add a comment |
I was wondering about the convergence of a sequence based on the convergence of the logarithm of the sequence. I can't seem to find anything online, and it seems to make intuitive sense, but I'm not entirely sure and am finding it difficult to show it.
To state the issue, if we have $log(a_n)$ converges, can it be shown that $a_n$ likewise converges to the value of $e$ raised to the limit?
Again, this seems somewhat intuitive, but I can't seem to figure out how to prove it effectively or find anything online. Any help would be appreciated.
real-analysis sequences-and-series logarithms
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
add a comment |
I was wondering about the convergence of a sequence based on the convergence of the logarithm of the sequence. I can't seem to find anything online, and it seems to make intuitive sense, but I'm not entirely sure and am finding it difficult to show it.
To state the issue, if we have $log(a_n)$ converges, can it be shown that $a_n$ likewise converges to the value of $e$ raised to the limit?
Again, this seems somewhat intuitive, but I can't seem to figure out how to prove it effectively or find anything online. Any help would be appreciated.
real-analysis sequences-and-series logarithms
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
I was wondering about the convergence of a sequence based on the convergence of the logarithm of the sequence. I can't seem to find anything online, and it seems to make intuitive sense, but I'm not entirely sure and am finding it difficult to show it.
To state the issue, if we have $log(a_n)$ converges, can it be shown that $a_n$ likewise converges to the value of $e$ raised to the limit?
Again, this seems somewhat intuitive, but I can't seem to figure out how to prove it effectively or find anything online. Any help would be appreciated.
real-analysis sequences-and-series logarithms
real-analysis sequences-and-series logarithms
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
edited Dec 11 '18 at 15:53
Davide Giraudo
125k16150260
125k16150260
asked Dec 11 '18 at 15:00
beeselmane
1135
asked Dec 11 '18 at 15:00
beeselmane
1135
asked Dec 11 '18 at 15:00
beeselmane
1135
1135
add a comment |
add a comment |
1 Answer
1
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oldest
votes
Hint: $exp$ is a continuous function.
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: $exp$ is a continuous function.
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
add a comment |
Hint: $exp$ is a continuous function.
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
add a comment |
Hint: $exp$ is a continuous function.
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
Hint: $exp$ is a continuous function.
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
answered Dec 11 '18 at 15:01
Robert Israel
318k23208457
318k23208457
add a comment |
add a comment |
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