Evaluating the continued fraction
0
How to evaluate $frac{1}{a_{1}+frac{1}{a_{2}+frac{1}{a_{3}+frac{1}{a_{4}+frac{1}{ddots}}}}}$, where $a_{j}=frac{1}{1}+frac{1}{2}+frac{1}{3}+dots+frac{1}{j}$?
That is,
How to evaluate
$frac{1}{left ( frac{1}{1} right )+frac{1}{left ( frac{3}{2} right)+frac{1}{left ( frac{11}{6} right )+frac{1}{left (
frac{25}{12}right )+frac{1}{ddots}}}}}$?
The value of the expression is $0.6606dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?
sequences-and-series summation continued-fractions harmonic-numbers
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
add a comment |
0
How to evaluate $frac{1}{a_{1}+frac{1}{a_{2}+frac{1}{a_{3}+frac{1}{a_{4}+frac{1}{ddots}}}}}$, where $a_{j}=frac{1}{1}+frac{1}{2}+frac{1}{3}+dots+frac{1}{j}$?
That is,
How to evaluate
$frac{1}{left ( frac{1}{1} right )+frac{1}{left ( frac{3}{2} right)+frac{1}{left ( frac{11}{6} right )+frac{1}{left (
frac{25}{12}right )+frac{1}{ddots}}}}}$?
The value of the expression is $0.6606dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?
sequences-and-series summation continued-fractions harmonic-numbers
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
4
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
4
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
1
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35
add a comment |
0
0
0
3
How to evaluate $frac{1}{a_{1}+frac{1}{a_{2}+frac{1}{a_{3}+frac{1}{a_{4}+frac{1}{ddots}}}}}$, where $a_{j}=frac{1}{1}+frac{1}{2}+frac{1}{3}+dots+frac{1}{j}$?
That is,
How to evaluate
$frac{1}{left ( frac{1}{1} right )+frac{1}{left ( frac{3}{2} right)+frac{1}{left ( frac{11}{6} right )+frac{1}{left (
frac{25}{12}right )+frac{1}{ddots}}}}}$?
The value of the expression is $0.6606dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?
sequences-and-series summation continued-fractions harmonic-numbers
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
How to evaluate $frac{1}{a_{1}+frac{1}{a_{2}+frac{1}{a_{3}+frac{1}{a_{4}+frac{1}{ddots}}}}}$, where $a_{j}=frac{1}{1}+frac{1}{2}+frac{1}{3}+dots+frac{1}{j}$?
That is,
How to evaluate
$frac{1}{left ( frac{1}{1} right )+frac{1}{left ( frac{3}{2} right)+frac{1}{left ( frac{11}{6} right )+frac{1}{left (
frac{25}{12}right )+frac{1}{ddots}}}}}$?
The value of the expression is $0.6606dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?
sequences-and-series summation continued-fractions harmonic-numbers
sequences-and-series summation continued-fractions harmonic-numbers
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
edited Dec 7 '18 at 19:28
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
asked Dec 7 '18 at 9:14
Hussain-Alqatari
2997
2997
4
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
4
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
1
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35
add a comment |
4
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
4
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
1
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35
4
4
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
4
4
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
1
1
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35
add a comment |
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4
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
– Peter
Dec 7 '18 at 9:20
@Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation.
– Hussain-Alqatari
Dec 7 '18 at 9:26
4
@Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form.
– Christoph
Dec 7 '18 at 9:31
1
Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior.
– Eevee Trainer
Dec 7 '18 at 9:33
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35