Evaluating the continued fraction












0












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
    $endgroup$
    – Peter
    Dec 7 '18 at 9:20










  • $begingroup$
    @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.
    $endgroup$
    – Hussain-Alqatari
    Dec 7 '18 at 9:26








  • 4




    $begingroup$
    @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.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:31








  • 1




    $begingroup$
    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.
    $endgroup$
    – Eevee Trainer
    Dec 7 '18 at 9:33












  • $begingroup$
    The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
    $endgroup$
    – Dr. Wolfgang Hintze
    Dec 8 '18 at 17:35
















0












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
    $endgroup$
    – Peter
    Dec 7 '18 at 9:20










  • $begingroup$
    @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.
    $endgroup$
    – Hussain-Alqatari
    Dec 7 '18 at 9:26








  • 4




    $begingroup$
    @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.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:31








  • 1




    $begingroup$
    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.
    $endgroup$
    – Eevee Trainer
    Dec 7 '18 at 9:33












  • $begingroup$
    The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
    $endgroup$
    – Dr. Wolfgang Hintze
    Dec 8 '18 at 17:35














0












0








0


3



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 19:28







Hussain-Alqatari

















asked Dec 7 '18 at 9:14









Hussain-AlqatariHussain-Alqatari

3187




3187








  • 4




    $begingroup$
    Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
    $endgroup$
    – Peter
    Dec 7 '18 at 9:20










  • $begingroup$
    @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.
    $endgroup$
    – Hussain-Alqatari
    Dec 7 '18 at 9:26








  • 4




    $begingroup$
    @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.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:31








  • 1




    $begingroup$
    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.
    $endgroup$
    – Eevee Trainer
    Dec 7 '18 at 9:33












  • $begingroup$
    The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
    $endgroup$
    – Dr. Wolfgang Hintze
    Dec 8 '18 at 17:35














  • 4




    $begingroup$
    Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
    $endgroup$
    – Peter
    Dec 7 '18 at 9:20










  • $begingroup$
    @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.
    $endgroup$
    – Hussain-Alqatari
    Dec 7 '18 at 9:26








  • 4




    $begingroup$
    @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.
    $endgroup$
    – Christoph
    Dec 7 '18 at 9:31








  • 1




    $begingroup$
    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.
    $endgroup$
    – Eevee Trainer
    Dec 7 '18 at 9:33












  • $begingroup$
    The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
    $endgroup$
    – Dr. Wolfgang Hintze
    Dec 8 '18 at 17:35








4




4




$begingroup$
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
$endgroup$
– Peter
Dec 7 '18 at 9:20




$begingroup$
Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation"
$endgroup$
– Peter
Dec 7 '18 at 9:20












$begingroup$
@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.
$endgroup$
– Hussain-Alqatari
Dec 7 '18 at 9:26






$begingroup$
@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.
$endgroup$
– Hussain-Alqatari
Dec 7 '18 at 9:26






4




4




$begingroup$
@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.
$endgroup$
– Christoph
Dec 7 '18 at 9:31






$begingroup$
@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.
$endgroup$
– Christoph
Dec 7 '18 at 9:31






1




1




$begingroup$
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.
$endgroup$
– Eevee Trainer
Dec 7 '18 at 9:33






$begingroup$
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.
$endgroup$
– Eevee Trainer
Dec 7 '18 at 9:33














$begingroup$
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
$endgroup$
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35




$begingroup$
The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954
$endgroup$
– Dr. Wolfgang Hintze
Dec 8 '18 at 17:35










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