Newton's Generalization of the Binomial Theorem and big o notation The Next CEO of Stack Overflowmaclaurin series and induction over binomial theoremGeneralization Of The Binomial TheoremTaylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?Using Newton's Generalized Binomial TheoremUse induction and Newton's binomial formula to show that $binomn0+binomn1+cdot+binomnn=2^n, forall nin mathbb N$Proof of Newton's Binomial ExpansionOn a connection between Newton's binomial theorem and general Leibniz rule using a new method.Newton's Generalization of the Binomial TheoremVerifying Newton's Binomial Series for $(1+x)^−2$Proof of Newton's Generalized Binomial Theorem (without Calculus)

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Newton's Generalization of the Binomial Theorem and big o notation



The Next CEO of Stack Overflowmaclaurin series and induction over binomial theoremGeneralization Of The Binomial TheoremTaylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?Using Newton's Generalized Binomial TheoremUse induction and Newton's binomial formula to show that $binomn0+binomn1+cdot+binomnn=2^n, forall nin mathbb N$Proof of Newton's Binomial ExpansionOn a connection between Newton's binomial theorem and general Leibniz rule using a new method.Newton's Generalization of the Binomial TheoremVerifying Newton's Binomial Series for $(1+x)^−2$Proof of Newton's Generalized Binomial Theorem (without Calculus)










0












$begingroup$


we know that :




(Newton's Generalization of the Binomial Theorem)



Let $x,y∈mathbbR$ where $0≤∣x∣<∣y∣$ and let $α∈mathbbR$. Then the expansion of the binomial $(x+y)^α$ is given by the infinite series



$$(x+y)^α=binomalpha0x^0y^α+binomalpha1x^1y^alpha -1+binomalpha2x^αy^alpha-2+...=∑_k=0^inftybinomalphakx^ky^α−k$$




Now let $y=1$



$$beginalign(x+1)^α&=underbracebinomalpha0x^0+cdots+binomalphanx^n+binomalphan+1x^n+1+cdots\
&=∑_k=0^inftybinomalphakx^k
endalign$$

So we have :
$$(x+1)^α-Big(underbracebinomalpha0x^0+cdots+binomalphanx^nBig)=x^n∑_k=n+1^inftybinomalphakx^k-n$$
And we know that :
$$x^n∑_k=n^inftybinomalphakx^k-n$$
where again the series converges for $0≤∣x∣<1$ .
Now how prove that :
$$x^n∑_k=n^inftybinomalphakx^k-n=O(x^n)$$










share|cite|improve this question









$endgroup$
















    0












    $begingroup$


    we know that :




    (Newton's Generalization of the Binomial Theorem)



    Let $x,y∈mathbbR$ where $0≤∣x∣<∣y∣$ and let $α∈mathbbR$. Then the expansion of the binomial $(x+y)^α$ is given by the infinite series



    $$(x+y)^α=binomalpha0x^0y^α+binomalpha1x^1y^alpha -1+binomalpha2x^αy^alpha-2+...=∑_k=0^inftybinomalphakx^ky^α−k$$




    Now let $y=1$



    $$beginalign(x+1)^α&=underbracebinomalpha0x^0+cdots+binomalphanx^n+binomalphan+1x^n+1+cdots\
    &=∑_k=0^inftybinomalphakx^k
    endalign$$

    So we have :
    $$(x+1)^α-Big(underbracebinomalpha0x^0+cdots+binomalphanx^nBig)=x^n∑_k=n+1^inftybinomalphakx^k-n$$
    And we know that :
    $$x^n∑_k=n^inftybinomalphakx^k-n$$
    where again the series converges for $0≤∣x∣<1$ .
    Now how prove that :
    $$x^n∑_k=n^inftybinomalphakx^k-n=O(x^n)$$










    share|cite|improve this question









    $endgroup$














      0












      0








      0





      $begingroup$


      we know that :




      (Newton's Generalization of the Binomial Theorem)



      Let $x,y∈mathbbR$ where $0≤∣x∣<∣y∣$ and let $α∈mathbbR$. Then the expansion of the binomial $(x+y)^α$ is given by the infinite series



      $$(x+y)^α=binomalpha0x^0y^α+binomalpha1x^1y^alpha -1+binomalpha2x^αy^alpha-2+...=∑_k=0^inftybinomalphakx^ky^α−k$$




      Now let $y=1$



      $$beginalign(x+1)^α&=underbracebinomalpha0x^0+cdots+binomalphanx^n+binomalphan+1x^n+1+cdots\
      &=∑_k=0^inftybinomalphakx^k
      endalign$$

      So we have :
      $$(x+1)^α-Big(underbracebinomalpha0x^0+cdots+binomalphanx^nBig)=x^n∑_k=n+1^inftybinomalphakx^k-n$$
      And we know that :
      $$x^n∑_k=n^inftybinomalphakx^k-n$$
      where again the series converges for $0≤∣x∣<1$ .
      Now how prove that :
      $$x^n∑_k=n^inftybinomalphakx^k-n=O(x^n)$$










      share|cite|improve this question









      $endgroup$




      we know that :




      (Newton's Generalization of the Binomial Theorem)



      Let $x,y∈mathbbR$ where $0≤∣x∣<∣y∣$ and let $α∈mathbbR$. Then the expansion of the binomial $(x+y)^α$ is given by the infinite series



      $$(x+y)^α=binomalpha0x^0y^α+binomalpha1x^1y^alpha -1+binomalpha2x^αy^alpha-2+...=∑_k=0^inftybinomalphakx^ky^α−k$$




      Now let $y=1$



      $$beginalign(x+1)^α&=underbracebinomalpha0x^0+cdots+binomalphanx^n+binomalphan+1x^n+1+cdots\
      &=∑_k=0^inftybinomalphakx^k
      endalign$$

      So we have :
      $$(x+1)^α-Big(underbracebinomalpha0x^0+cdots+binomalphanx^nBig)=x^n∑_k=n+1^inftybinomalphakx^k-n$$
      And we know that :
      $$x^n∑_k=n^inftybinomalphakx^k-n$$
      where again the series converges for $0≤∣x∣<1$ .
      Now how prove that :
      $$x^n∑_k=n^inftybinomalphakx^k-n=O(x^n)$$







      calculus algebra-precalculus






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Mar 19 at 12:12









      Almot1960Almot1960

      2,325825




      2,325825




















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