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Math Help - √2 as a sum of infinite series

  1. #1
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    √2 as a sum of infinite series

    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.
    Last edited by ThodorisK; February 6th 2011 at 12:06 AM.
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    Quote Originally Posted by ThodorisK View Post
    I cant' find any. And some terms of the continued fraction e.g. the fourth term 1+(5/12), are greater than √2.
    Please explain the question.
    Do you simply a series the sum of which is \sqrt2~?
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    \displaystyle \sum_{k=0}^\infty \frac{(-1)^k(2k)!}{(1-2k)(k!)^2(4^k)}.
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    Quote Originally Posted by ThodorisK View Post
    I cant' find any.

    And some terms of the continued fraction e.g. the fourth term 1+(5/12), are greater than √2.
    1. Use the function f(x)=\sqrt{x}

    2. Develop f in a Taylor serie:

    f(a+h)=f(a)+\frac h{1!} f'(a) + \frac{h^2}{2!} f''(a) + ... + \frac{h^{n-1}}{(n-1)!} f^{(n-1)}(a) + R_n

    wit a = 1 and h = 1
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    \displaystyle \sqrt{2} = 2\ \cos \frac{\pi}{4}} = 2\ \sum_{n=0}^{\infty} (-1)^{n} \frac{(\frac{\pi}{4})^{2n}}{(2n)!}
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    I suspect the OP wants to express \sqrt{2} as a continued fraction. But until clarification of the question is provided, there is no point stabbing in the dark at possbile answers.
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    Hello, ThodorisK!

    If you want a continued fraction for \sqrt{2}, here's one . . .


    \displaystyle \sqrt{2} \;=\;1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \hdots}}}}}

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    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.
    Last edited by ThodorisK; February 6th 2011 at 02:32 AM.
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    Quote Originally Posted by ThodorisK View Post
    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.

    Also, I want it to contain numbers only. Or directions of how the numbers are combined, e.g. not something with π in it.

    Your last words seem to imply that you're not really sure either of what you

    want or else you're not quite sure what you were asked to do: the "something with n in it" is numbers in

    the examples you've been given, and the n only serves as counting index.

    A simple way to achieve what you want is:

    \displaystyle{1=\sum\limits^\infty_{n=1}\left(\fra  c{1}{2}\right)^n\Longrightarrow \sqrt{2}=\sum\limits^\infty_{n=1}\frac{1}{2^{n-1/2}}=\frac{1}{\sqrt{2}}+\frac{1}{2\sqrt{2}}+\frac{1  }{4\sqrt{2}}+\ldots}

    Tonio
    Last edited by tonio; February 6th 2011 at 01:04 AM.
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    Do you mean an infinite sequence of sums ?

    \displaystyle\ a_{n+1}=\frac{a_n}{2}+\frac{1}{a_n},\;\;\;\;a_0=1

    gives the sequence

    \displaystyle\ 1,\;\;\;\frac{1}{2}+1=\frac{3}{2},\;\;\;\frac{3}{4  }+\frac{2}{3}=\frac{17}{12}=1+\frac{5}{12},.......  ........

    though you say that should be the 4th term...
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  11. #11
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    Quote Originally Posted by ThodorisK View Post
    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.
    We are still in the dark.

    State the question exactly as it is written in your book (or wherever you got it from).
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    Let's say that I want something like the 1+2(1/8)+4(1/8)^2+8(1/8)^2+...of the area of the parabola
    Geometric series - Wikipedia, the free encyclopedia
    where instead of the area of the parabola, it is the side √2 of a triangle with sides 1,1,√2. So, none of the terms of the sum can be greater than √2. What I mean by "the terms of the sum" is for example regarding the continued fraction:

    first term = 1
    second term = 1+1/2
    third term = 1+(1/(2+(1/2))=1+(2/5)
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    Binomial theorem

    Quote Originally Posted by ThodorisK View Post
    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.
    I seem to recall that you can express \sqrt{2} as a series using the binomial theorem. What I can do for you is when I get home, I can look it up and get back to you tomorrow on it.
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    Square_root_of_2

    scroll down to Series and product representations
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    Quote Originally Posted by ThodorisK View Post
    I dont want THE continued fraction because it's not formed as a sum of infinite series. Also, when I try to make it as such, its 2nd, 4th, 6th and so on terms are greater than √2, e.g. the fourth term is 1+(5/12), so I guess it's not what I want, but I might be wrong.
    Do not modify your original post in response to something in a reply, but post a new message quoting what you are responding to and your response. That way the thread will make sense and can be followed by new viewers.

    CB
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