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Help with Taylor Series

  1. Feb 27, 2015 #1
    $$f(a + x) = \sum_{k=0}^∞ \frac{f^{(k)}(a) x^k}{k!}$$

    Usually written as:

    $$f(t) = \sum_{k=0}^∞ \frac{f^{(k)}(a) (t-a)^k}{k!}$$

    Where ##t = a + x##
    Is the taylor expansion supposed to give the same result for all ##a##? The reason this confuses me is because this seems to suggest that ##f(1 + x) = f(4 + x) = f(π + x)## and so on, which is usually not the case. Where did I go wrong?
     
  2. jcsd
  3. Feb 27, 2015 #2

    Stephen Tashi

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    Yes, in the sense that for a given [itex] t [/itex]

    [itex] f(t) = \sum_{k=0}^\infty \frac{f^{(k)}(a_1) (t-a_1)^k}{k!} = \sum_{k=0}^\infty \frac{f^{(k)}(a_2)(t-a_2)^k}{k!} [/itex]

    when both series converge.


    In the first version you listed for Taylor series, to say that the equation holds for different [itex] a_1, a_2 [/itex] does not imply [itex] f(a_1 + x) = f(a_2 + x) [/itex] since, in general, [itex] f^{(k)}(a_1) \ne f^{(k)}(a_2) [/itex]. In the second version you listed, different values of [itex] a_1, a_2 [/itex] do not imply different values of [itex] t [/itex].
     
  4. Feb 27, 2015 #3
    So the variable ##x## must also change for ##t## to remain well-defined.
     
  5. Feb 27, 2015 #4

    Stephen Tashi

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    The meaning of that statement isn't entirely clear, but I'm tempted to say yes.
     
  6. Feb 27, 2015 #5
    In order to maintain the same meaning of ##t## (and ##f(t)##) while changing the value of ##a##, we must change the value of ##x##.
     
  7. Feb 27, 2015 #6

    Stephen Tashi

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    Yes, if we assert [itex] t = a + x [/itex].
     
  8. Feb 28, 2015 #7
    If we were to swap the positions of all ##a##'s and ##x##'s in the expansion; that is: ##f(x) + f'(x)a + \frac{f''(x)a^2}{2!} + ....##, would we get the same result? I understand that our goal is to write a power series where the powers are constantly increasing on a variable, not a constant. But I'm curious, would this give the same result?
     
  9. Feb 28, 2015 #8

    Stephen Tashi

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    A logician could object to calling such a change in the formula "the same result" since logician wants it made clear which symbols are variables (in some scope) and which are constants (in some scope). Technically, a symbol like "x", should appear with the scope of a quantifier (like "for each") when we write a mathematical statement. People who do calculus are careful about quantifying their variables when they do epsilon-delta proofs (e.g. for each epsilon > 0, there exists a delta ...such that...), but in other aspects of calculus they are careless. You can talk about "Taylor's Formula" as a string of symbols. To ask a precise mathematical question about its meaning you need to first state "Taylor's Theorem" as a theorem. See which quantifiers apply to which variables. Then you can ask whether the symbols representing the variables can be swapped.
     
  10. Feb 28, 2015 #9
    (a_n)_1^infty
     
  11. Feb 28, 2015 #10
    ({a_n})_1^ \infty
     
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