Reversing a MacLaurin/Taylor Expansion

Discussion in 'General Math' started by Alastair, Mar 19, 2005.

  1. Alastair

    Alastair Guest

    Numerous functions can be expressed as an infinite series via the
    MacLaurin method of expansion;

    f(x)= f(0) + xf'(0) + [(x^2)/2!]f''(0) + ... + [(x^r)/r!]f^r(0) + ...
    =Sum_{r = 0 to infinity}[((x^r)/r!)*f^r(0)]

    {Where f^r(0) is the nth derivative of f(x) at x = 0}

    For example, e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

    My question is, can you take a known infinite series and reverse the
    process, to produce a function which does not involve an infinite
    summation?

    In this case, it would be showing/proving that

    1 + x + (x^2)/2! + (x^3)/3! + ... = e^x

    I'm simply curious on this one, but is it possible?

    Thanks already =)
     
    Alastair, Mar 19, 2005
    #1
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  2. Alastair

    Jim Spriggs Guest

    How would you do this other than by recognizing that an infinite series
    was known by some name already? Note that there are far more infinite
    series (and even far more MacLaurin expansions) than there are names in
    use.

    You could do it trivially by saying "I name <some infinite series> to be
    <some name of your choice>."

    Strictly speaking functions neither involve, nor fail to involve,
    infinite summation. A function is just a rule for associating an
    element of a set (say a real number) with the element of the same set or
    another set (say another real number). The "formula" (infinite
    summation or otherwise) is not the function, though in the past it was
    thought that all functions must be given by formulae.
    If e^x was defined to mean, let's say, lim_{n-->oo}(1 + x/n)^n, then one
    might prove that that limit is 1 + x + (x^2)/2! + (x^3)/3! + ... . Is
    that what you mean?
     
    Jim Spriggs, Mar 19, 2005
    #2
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  3. Alastair

    Alastair Guest

    My mistake. =)
    My question earlier may not have been entirely clear, so I'll try
    again.

    This method seems to me like you will be proving it by taking e^x and
    turning it into an infinite series, and showing that the two series are
    the same. Imagine that you're given some infinite series, and you have
    no idea how to express it without using infinite summation. For
    example, if I presented you with the series such as "1 + x + (x^2)/2! +
    (x^3)/3! + ... " and asked you to express it as a finite number of
    terms, would there be some method (other than quoting standard
    results!) of showing that the series was equivalent to some much
    shorter expression? Or if the series was *not* equivalent to some
    shorter expression, could this be proved?
     
    Alastair, Mar 20, 2005
    #3
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