Sum of logs $log x + loglog x +…$
$begingroup$
Is there a reduction for this infinite sum?
$$log x + loglog x + logloglog x +... = ?$$
for all $x > 0$?
summation
edited Jan 24 at 20:46
T. Bongers
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
$endgroup$
add a comment |
$begingroup$
Is there a reduction for this infinite sum?
$$log x + loglog x + logloglog x +... = ?$$
for all $x > 0$?
summation
edited Jan 24 at 20:46
T. Bongers
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
$endgroup$
2
$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47
add a comment |
$begingroup$
Is there a reduction for this infinite sum?
$$log x + loglog x + logloglog x +... = ?$$
for all $x > 0$?
summation
edited Jan 24 at 20:46
T. Bongers
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
$endgroup$
Is there a reduction for this infinite sum?
$$log x + loglog x + logloglog x +... = ?$$
for all $x > 0$?
summation
summation
edited Jan 24 at 20:46
T. Bongers
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
edited Jan 24 at 20:46
T. Bongers
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
edited Jan 24 at 20:46
T. Bongers
23.5k54762
edited Jan 24 at 20:46
T. Bongers
23.5k54762
edited Jan 24 at 20:46
T. Bongers
23.5k54762
23.5k54762
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
asked Jan 24 at 20:42
Matt DeleeuwMatt Deleeuw
113
113
2
$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47
add a comment |
2
$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47
2
2
$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47
$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47
add a comment |
1 Answer
1
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$begingroup$
The sum diverges with any reasonable interpretation of the logarithms. After all, $log log cdots log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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votes
$begingroup$
The sum diverges with any reasonable interpretation of the logarithms. After all, $log log cdots log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
$endgroup$
add a comment |
$begingroup$
The sum diverges with any reasonable interpretation of the logarithms. After all, $log log cdots log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
$endgroup$
add a comment |
$begingroup$
The sum diverges with any reasonable interpretation of the logarithms. After all, $log log cdots log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
$endgroup$
The sum diverges with any reasonable interpretation of the logarithms. After all, $log log cdots log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
answered Jan 24 at 20:44
T. BongersT. Bongers
23.5k54762
23.5k54762
add a comment |
add a comment |
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$begingroup$
Possible duplicate of In which interval (domain) does the $sum_{n=1}^{infty}log^n(1+x)$ converge absolutely?
$endgroup$
– Tito Eliatron
Jan 24 at 20:47