Summation using binomial theorem

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Find the sum $$sum_{x=1}^{m} Big(frac{2^x-1}{2^x}Big)^n$$ where $n$ belongs to natural number.










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  • $begingroup$
    Is anything known about $m$ and $n$ aside from them being natural numbers?
    $endgroup$
    – Aleksejs Fomins
    Jan 21 at 8:38










  • $begingroup$
    @AleksejsFomins No Sir.
    $endgroup$
    – user574937
    Jan 23 at 9:23










  • $begingroup$
    It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
    $endgroup$
    – Aleksejs Fomins
    Jan 23 at 10:03
















0












$begingroup$


Find the sum $$sum_{x=1}^{m} Big(frac{2^x-1}{2^x}Big)^n$$ where $n$ belongs to natural number.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Is anything known about $m$ and $n$ aside from them being natural numbers?
    $endgroup$
    – Aleksejs Fomins
    Jan 21 at 8:38










  • $begingroup$
    @AleksejsFomins No Sir.
    $endgroup$
    – user574937
    Jan 23 at 9:23










  • $begingroup$
    It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
    $endgroup$
    – Aleksejs Fomins
    Jan 23 at 10:03














0












0








0


1



$begingroup$


Find the sum $$sum_{x=1}^{m} Big(frac{2^x-1}{2^x}Big)^n$$ where $n$ belongs to natural number.










share|cite|improve this question











$endgroup$




Find the sum $$sum_{x=1}^{m} Big(frac{2^x-1}{2^x}Big)^n$$ where $n$ belongs to natural number.







summation binomial-coefficients binomial-theorem






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share|cite|improve this question













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share|cite|improve this question








edited Jan 21 at 7:54









Shubham Johri

5,187717




5,187717










asked Jan 21 at 7:27









user574937user574937

384




384












  • $begingroup$
    Is anything known about $m$ and $n$ aside from them being natural numbers?
    $endgroup$
    – Aleksejs Fomins
    Jan 21 at 8:38










  • $begingroup$
    @AleksejsFomins No Sir.
    $endgroup$
    – user574937
    Jan 23 at 9:23










  • $begingroup$
    It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
    $endgroup$
    – Aleksejs Fomins
    Jan 23 at 10:03


















  • $begingroup$
    Is anything known about $m$ and $n$ aside from them being natural numbers?
    $endgroup$
    – Aleksejs Fomins
    Jan 21 at 8:38










  • $begingroup$
    @AleksejsFomins No Sir.
    $endgroup$
    – user574937
    Jan 23 at 9:23










  • $begingroup$
    It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
    $endgroup$
    – Aleksejs Fomins
    Jan 23 at 10:03
















$begingroup$
Is anything known about $m$ and $n$ aside from them being natural numbers?
$endgroup$
– Aleksejs Fomins
Jan 21 at 8:38




$begingroup$
Is anything known about $m$ and $n$ aside from them being natural numbers?
$endgroup$
– Aleksejs Fomins
Jan 21 at 8:38












$begingroup$
@AleksejsFomins No Sir.
$endgroup$
– user574937
Jan 23 at 9:23




$begingroup$
@AleksejsFomins No Sir.
$endgroup$
– user574937
Jan 23 at 9:23












$begingroup$
It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
$endgroup$
– Aleksejs Fomins
Jan 23 at 10:03




$begingroup$
It does not appear obvious that there is a simple closed-form function corresponding to the above sum. Why do you believe that the above sum can be found?
$endgroup$
– Aleksejs Fomins
Jan 23 at 10:03










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