Converting Sigma notation...

  1. Converting Sigma notation....

    1. The problem statement, all variables and given/known data
    (Not a homework question)
    Hi!!
    I have been encountering problems in Binomial Theorem which includes converting the sigma notation.
    Like [tex]\sum_{k=0}^n \frac{n!}{(n-k)!k!} a^kb^{n-k}=(a+b)^n[/tex]

    I got many questions in my exam of this type with four options.
    One of them was:-
    [tex]\sum_{k=1}^{n} {}^nC_k.3^k[/tex]
    I substituted the value of n and was able to figure out the correct option.
    But as i said there were many questions, so it took a lot of time.
    Is there any easier way to do that? :confused:

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Jun 28, 2011
  2. jcsd
  3. Mentallic

    Mentallic 3,691
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    Re: Converting Sigma notation....

    The binomial theorem is:

    [tex]\sum_{k=0}^n \frac{n!}{(n-k)!k!}a^kb^{n-k}=(a+b)^n[/tex]

    So this would assume that a=b=1. I'm not sure if this was a question or you just incorrectly wrote down the binomial expansion though...

    I'm still not quite getting it. That is just an expression, but what is the actual question?
     
  4. Re: Converting Sigma notation....

    Sorry!! I incorrectly wrote down the binomial expansion.

    Like the binomial expansion can be written to (a+b)n, i want to write the given expression in the form as we compress the sigma notation to (a+b)n.
    (Would you please tell me how to make the "n" before "C" in Superscript?)

    I hope you get it now.:smile:
     
  5. Mentallic

    Mentallic 3,691
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    Re: Converting Sigma notation....

    Oh ok I see, well then since we have the binomial expansion involves both ak and bn-k (so in other words, just two values, each being raised to some power) and you're trying to find

    [tex]\sum_{k=1}^n ^nC_k.3^k[/tex] to create a superscript in [itex]\LaTeX[/itex] just use ^ (and add {} for multiple characters) before it :wink:

    Notice that we can see a=3, but b isn't present. In fact, it's just hidden as b=1 because

    [tex]\sum_{k=1}^n ^nC_k.3^k1^{n-k}[/tex]

    is exactly the same thing. So our final answer would be

    [tex]\sum_{k=0}^n ^nC_k.3^k-^nC_03^0=(3+1)^n-1=4^n-1[/tex]

    EDIT: and it seems that latex has changed, once again... Man I'm getting annoyed with it. Give me a second and I'll figure it out.
     
  6. vela

    vela 12,561
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    Re: Converting Sigma notation....

    You can use {^n}C_k or {}^n C_k.
     
  7. Re: Converting Sigma notation....

    Thanks Mentallic but i don't understand from where you got
    [tex]-^nC_03^0[/tex]:confused:


    Thanks vela, it worked :smile:
     
  8. I like Serena

    I like Serena 6,193
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    Re: Converting Sigma notation....

    Hi Pranav-Arora! :smile:

    Seeing that Mentallic and vela are not around, I'll answer your question.

    It's part of the sigma notation and its implications.
    In particular this is about the boundaries of the sum, which in your case is starting with k=1.
    What you have is:

    [tex]\begin{eqnarray}
    \sum_{k=1}^n {^n}C_k \cdot 3^k
    &=&{^n}C_1 \cdot 3^1 + {^n}C_2 \cdot 3^2 + \dotsb + {^n}C_n \cdot 3^n\\
    &=&({^n}C_0 \cdot 3^0 + {^n}C_1 \cdot 3^1 + {^n}C_2 \cdot 3^2 + \dotsb + {^n}C_n \cdot 3^n) - {^n}C_0 \cdot 3^0 \\
    &=&(\sum_{k=0}^n {^n}C_k \cdot 3^k) - {^n}C_0 \cdot 3^0
    \end{eqnarray}[/tex]

    The way to change the boundaries is always the same.
    Your write out the sum in its terms, change what you want to change, and change it back again into sigma notation.
    Note that the sigma notation is only a shorthand notation. Don't think of it as something magical that has its own rules - it hasn't. It's just shorthand.
     
  9. Re: Converting Sigma notation....

    Thanks for your reply I like Serena. :smile:
    But what i would do if a question appears like this:-

    [tex]\sum^n_{k=0} (2k+1) {}^nC_k[/tex]. :confused:
     
  10. I like Serena

    I like Serena 6,193
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    Re: Converting Sigma notation....

    Ah, this one is a bit more difficult.
    The method I know is to define a function of x and integrate it.
    That is:
    [tex]s(x) = \sum^n_{k=0} {}^nC_k (2k+1) x^{2k}[/tex]
    The result you're looking for in this case is s(1).

    If you integrate it, you should find a form that looks more like your previous problem.
    You can rewrite that without the sigma and binomium.
    Afterward you differentiate again.
    And finally you fill in the value 1.

    Care to try? :smile:
     
  11. Re: Converting Sigma notation....

    How you get x2k? :confused:
     
  12. Mentallic

    Mentallic 3,691
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    Re: Converting Sigma notation....

    Good thing you did considering I was miles away from any internet connections (locked away in my room, sleeping) :smile:

    What happens if you integrate x2k?
     
  13. vela

    vela 12,561
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    Re: Converting Sigma notation....

    Is this problem from a pre-calculus class since you posted in the precalc forum? If so, did you learn any identities involving binomial coefficients?
     
  14. Re: Converting Sigma notation....

    Maybe
    [tex]\frac{2(x^{2k+1})}{2k+1}[/tex]
     
  15. Re: Converting Sigma notation....

    Yep, i have learned some identities involving binomial coefficients but i didn't knew that to solve this question we have to perform integration and differetiation. :wink:
     
  16. Mentallic

    Mentallic 3,691
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    Re: Converting Sigma notation....

    Not quite. Take the derivative of that to see where you went wrong.

    They might have given you a formula to use?
     
  17. Re: Converting Sigma notation....

    I took the derivative i again got x2k. What is your answer when you integrate x2k? :confused:

    I have only one formula which involves integration within limits. But i only know how to integrate without limits. I have never solved questions which involve integration within limits.
    Here's the formula:-

    [tex](1+x)^n=C_0+C_1x+C_2x^2+........+C_kx^k+.......+C_nx^n...[/tex]

    Integrating the above equation with respect to x between limits 0 to 1, (I don't understand what it is :confused:) we get,

    [tex](\frac {(1+x)^{n+1}}{n+1})_0^1=(C_0x+C_1\frac{x^2}{2}+C_2\frac{x^3}{3}+.....C_n\frac{x^{n+1}}{n+1})_0^1[/tex]

    [tex]=C_0+\frac{C_1}{2}+\frac{C_2}{3}.....\frac{C_n}{n+1}=\frac{2^{2n+1}-1}{n+1}[/tex]
     
  18. Mentallic

    Mentallic 3,691
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    Re: Converting Sigma notation....

    The integral of xn is [tex]\frac{x^{n+1}}{n+1}[/tex] where n is a constant. Since k is just some constant, the same rule applies. It looks as though you're treating k as if it's a variable.


    I have only one formula which involves integration within limits. But i only know how to integrate without limits. I have never solved questions which involve integration within limits.
    Here's the formula:-

    [tex](1+x)^n=C_0+C_1x+C_2x^2+........+C_kx^k+.......+C_nx^n...[/tex][/quote]

    If that is the formula, then when evaluating this at 1, we would have

    [tex](1+1)^n=2^n=C_0+C_1+C_2+...+C_k+...+C_n[/tex] correct?

    and evaluating it at 0

    [tex](1+0)^n=1^n=1=C_0+C_1\cdot 0+C_2\cdot 0+...=1[/tex]

    What don't you understand about it? If two sides are equal, then the integral of both sides will be equal (disregarding the constant of integration).

    So can you apply this to your question now?
     
  19. Re: Converting Sigma notation....

    How would i apply this to my question?

    I like Serena said that define a function of x and integrate it but i still don't get how he got x2k? :confused:
     
  20. I like Serena

    I like Serena 6,193
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    Re: Converting Sigma notation....

    I defined an arbitrary function s(x) that looks a bit like your problem with the special property that if you substitute x=1, it is identical to your problem.

    It's a trick to solve your problem. :smile:

    The choice of the power 2k was inspired so that the factor (2k+1) would disappear during integration.
     
  21. Re: Converting Sigma notation....

    Ok i got it!! :smile:
    But how i would integrate the expression. It involves sigma notation and i have never done integration of any expression which involves sigma notation. :frown:
     
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