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Prove $sum_i=1^nfrac1sqrtilefrac1sqrtn!prod_i=2^n(sqrti-1)+2sum_i=2^nfrac1sqrti$ using Weierstrass inequality


Proving the Power Mean Inequality using Chebyshev's sum inequalityHow prove this inequality $fracasqrtb^2+b+1+fracbsqrtc^2+c+1+fraccsqrta^2+a+1gesqrt3$Generalized Bernoulli's inequality of the form $frac11-sum_i=1^nx_igeqprod_i=1^n(1+x_i)$How prove Reversing the Arithmetic mean – Geometric mean inequality?Prove that $ sqrtfracab+3 +sqrtfracbc+3 +sqrtfracca+3 leq frac32 $How to prove this inequality? ThanksProve inequality using Jensen's inequalityInequality : $ sum_cyc fraca+bsqrta+2c geq 2 sqrta+b+c$How to prove the Weierstrass product inequality using recurrency?













0












$begingroup$



Use Weierstrass's inequality to prove that
$$sum_i=1^n frac1sqrtile frac1sqrtn! prod_i = 2^n (sqrti-1) + 2 left(sum_i = 2^n frac1sqrtiright).$$




Using Weierstrass's inequality, I get



$$1 + sum_i=1^n frac1sqrti le frac1sqrtn! left(prod_i = 1^n (1 + sqrti)right)$$










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$endgroup$











  • $begingroup$
    The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
    $endgroup$
    – Viktor Glombik
    Mar 21 at 18:09
















0












$begingroup$



Use Weierstrass's inequality to prove that
$$sum_i=1^n frac1sqrtile frac1sqrtn! prod_i = 2^n (sqrti-1) + 2 left(sum_i = 2^n frac1sqrtiright).$$




Using Weierstrass's inequality, I get



$$1 + sum_i=1^n frac1sqrti le frac1sqrtn! left(prod_i = 1^n (1 + sqrti)right)$$










share|cite|improve this question











$endgroup$











  • $begingroup$
    The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
    $endgroup$
    – Viktor Glombik
    Mar 21 at 18:09














0












0








0





$begingroup$



Use Weierstrass's inequality to prove that
$$sum_i=1^n frac1sqrtile frac1sqrtn! prod_i = 2^n (sqrti-1) + 2 left(sum_i = 2^n frac1sqrtiright).$$




Using Weierstrass's inequality, I get



$$1 + sum_i=1^n frac1sqrti le frac1sqrtn! left(prod_i = 1^n (1 + sqrti)right)$$










share|cite|improve this question











$endgroup$





Use Weierstrass's inequality to prove that
$$sum_i=1^n frac1sqrtile frac1sqrtn! prod_i = 2^n (sqrti-1) + 2 left(sum_i = 2^n frac1sqrtiright).$$




Using Weierstrass's inequality, I get



$$1 + sum_i=1^n frac1sqrti le frac1sqrtn! left(prod_i = 1^n (1 + sqrti)right)$$







real-analysis inequality






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 21 at 18:04









Viktor Glombik

1,3122628




1,3122628










asked Mar 21 at 17:29









JekiJeki

53




53











  • $begingroup$
    The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
    $endgroup$
    – Viktor Glombik
    Mar 21 at 18:09

















  • $begingroup$
    The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
    $endgroup$
    – Viktor Glombik
    Mar 21 at 18:09
















$begingroup$
The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
$endgroup$
– Viktor Glombik
Mar 21 at 18:09





$begingroup$
The very same question has already been asked here, so please use approach0 to search before you ask, to see if your question (likely a homework problem if two people ask it in such a short time frame) has been ask before.
$endgroup$
– Viktor Glombik
Mar 21 at 18:09











1 Answer
1






active

oldest

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0












$begingroup$

Hint:
Since
$$
sum_k = 1^n frac1sqrtk = 1 + sum_k = 2^n frac1sqrtk,
$$

we can rewrite the inequality, which has to be shown (by subtracting $2 sum_k = 2^n frac1sqrtk$ from both sides) as
$$
1 - sum_k = 2^n frac1sqrtk
le frac1sqrtn! prod_k = 2^n (sqrtk-1).
$$

By the Weierstrass inequality (since $frac1sqrtk in (0,1]$ for all $k in mathbbN_>0$) we have
$$
1 - sum_k = 2^n frac1sqrtk
le prod_k = 2^n left(1 - frac1sqrtkright)
= prod_k = 2^n fracsqrtk - 1sqrtk
= frac1sqrtn! prod_k = 2^n (sqrtk-1),
le frac1sqrtn! prod_k = 2^n sqrtk - 1
$$

where the last equality is achieved by pulling out the denominator out of the product and the last inequality is using the fact that $sqrtk-1 le sqrtk - 1$ for all $k in mathbbN_>1$, which you can easily verify either by induction or by drawing a graph.






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    1 Answer
    1






    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    Hint:
    Since
    $$
    sum_k = 1^n frac1sqrtk = 1 + sum_k = 2^n frac1sqrtk,
    $$

    we can rewrite the inequality, which has to be shown (by subtracting $2 sum_k = 2^n frac1sqrtk$ from both sides) as
    $$
    1 - sum_k = 2^n frac1sqrtk
    le frac1sqrtn! prod_k = 2^n (sqrtk-1).
    $$

    By the Weierstrass inequality (since $frac1sqrtk in (0,1]$ for all $k in mathbbN_>0$) we have
    $$
    1 - sum_k = 2^n frac1sqrtk
    le prod_k = 2^n left(1 - frac1sqrtkright)
    = prod_k = 2^n fracsqrtk - 1sqrtk
    = frac1sqrtn! prod_k = 2^n (sqrtk-1),
    le frac1sqrtn! prod_k = 2^n sqrtk - 1
    $$

    where the last equality is achieved by pulling out the denominator out of the product and the last inequality is using the fact that $sqrtk-1 le sqrtk - 1$ for all $k in mathbbN_>1$, which you can easily verify either by induction or by drawing a graph.






    share|cite|improve this answer











    $endgroup$

















      0












      $begingroup$

      Hint:
      Since
      $$
      sum_k = 1^n frac1sqrtk = 1 + sum_k = 2^n frac1sqrtk,
      $$

      we can rewrite the inequality, which has to be shown (by subtracting $2 sum_k = 2^n frac1sqrtk$ from both sides) as
      $$
      1 - sum_k = 2^n frac1sqrtk
      le frac1sqrtn! prod_k = 2^n (sqrtk-1).
      $$

      By the Weierstrass inequality (since $frac1sqrtk in (0,1]$ for all $k in mathbbN_>0$) we have
      $$
      1 - sum_k = 2^n frac1sqrtk
      le prod_k = 2^n left(1 - frac1sqrtkright)
      = prod_k = 2^n fracsqrtk - 1sqrtk
      = frac1sqrtn! prod_k = 2^n (sqrtk-1),
      le frac1sqrtn! prod_k = 2^n sqrtk - 1
      $$

      where the last equality is achieved by pulling out the denominator out of the product and the last inequality is using the fact that $sqrtk-1 le sqrtk - 1$ for all $k in mathbbN_>1$, which you can easily verify either by induction or by drawing a graph.






      share|cite|improve this answer











      $endgroup$















        0












        0








        0





        $begingroup$

        Hint:
        Since
        $$
        sum_k = 1^n frac1sqrtk = 1 + sum_k = 2^n frac1sqrtk,
        $$

        we can rewrite the inequality, which has to be shown (by subtracting $2 sum_k = 2^n frac1sqrtk$ from both sides) as
        $$
        1 - sum_k = 2^n frac1sqrtk
        le frac1sqrtn! prod_k = 2^n (sqrtk-1).
        $$

        By the Weierstrass inequality (since $frac1sqrtk in (0,1]$ for all $k in mathbbN_>0$) we have
        $$
        1 - sum_k = 2^n frac1sqrtk
        le prod_k = 2^n left(1 - frac1sqrtkright)
        = prod_k = 2^n fracsqrtk - 1sqrtk
        = frac1sqrtn! prod_k = 2^n (sqrtk-1),
        le frac1sqrtn! prod_k = 2^n sqrtk - 1
        $$

        where the last equality is achieved by pulling out the denominator out of the product and the last inequality is using the fact that $sqrtk-1 le sqrtk - 1$ for all $k in mathbbN_>1$, which you can easily verify either by induction or by drawing a graph.






        share|cite|improve this answer











        $endgroup$



        Hint:
        Since
        $$
        sum_k = 1^n frac1sqrtk = 1 + sum_k = 2^n frac1sqrtk,
        $$

        we can rewrite the inequality, which has to be shown (by subtracting $2 sum_k = 2^n frac1sqrtk$ from both sides) as
        $$
        1 - sum_k = 2^n frac1sqrtk
        le frac1sqrtn! prod_k = 2^n (sqrtk-1).
        $$

        By the Weierstrass inequality (since $frac1sqrtk in (0,1]$ for all $k in mathbbN_>0$) we have
        $$
        1 - sum_k = 2^n frac1sqrtk
        le prod_k = 2^n left(1 - frac1sqrtkright)
        = prod_k = 2^n fracsqrtk - 1sqrtk
        = frac1sqrtn! prod_k = 2^n (sqrtk-1),
        le frac1sqrtn! prod_k = 2^n sqrtk - 1
        $$

        where the last equality is achieved by pulling out the denominator out of the product and the last inequality is using the fact that $sqrtk-1 le sqrtk - 1$ for all $k in mathbbN_>1$, which you can easily verify either by induction or by drawing a graph.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 21 at 17:57

























        answered Mar 21 at 17:49









        Viktor GlombikViktor Glombik

        1,3122628




        1,3122628



























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