- #1
v3nture
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This isn't a homework problem, a classmate asked for a challenging proof to try and do and this was the one we were given. We started by trying to derive some rules from un-integratable functions but realized that that would take a long time and a lot of work. After some thinking we came up with the idea of using Taylor Series and proof by definition to prove it. My question I guess is does that work? Here's the proof we came up with:
Let the set of all integrable functions be call V. All integrable functions can be expressed as a Taylor Series, all integrable functions can be defined as Taylor Series, when can then be integrated.
Let [itex]\oplus[/itex] be an operation defined by:
Let u, v, w be integrable functions defined as a Taylor Series [itex]\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}[/itex]
[tex]
u = \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}\\
v = \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n}\\
w = \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n}\\
1. u \oplus v = v \oplus u: \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} = \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}\\
\begin{align} 2. u \oplus (v \oplus w) = w \oplus (u \oplus w&): \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus (\sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n})\\ &= \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n} \oplus (\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n})\\ &= \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \end{align}\\
4. u \oplus -u = 0: \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} + -\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} = 0 \\
\text{Closed Under Addition} \oplus \text{, closed as:} \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}}\\
\text{Let } \odot \text{ be an operator defined by:} \mathbf{c} \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\ \text{ where } \mathbf{c} \text{ is a constant }
\forall u, v\epsilon \mathbf{V} \text{ and c,d}\epsilon\mathbf{R}\\
\mathbf{1) } \mathbf{ c} \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{c}\\
\mathbf{2) } \mathbf{ c } \odot (\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \sum_{\text{n=0}}^{\infty} \frac{g^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}) = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{g^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\mathbf{3) } \mathbf(c+d) \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \mathbf{d}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\mathbf{4) } \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{c}) \odot \mathbf{d} = \mathbf{c} \odot (\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{d})\\
\mathbf{5) } 1 \times \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\text{Scalar Multiplication } \odot \text{ closed as: } \mathbf{c } \odot \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}} = \int{\mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}} = \mathbf{c} \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}}\\ \\
\text{So, if the properties of } \oplus \text{ and } \odot \text{ hold under V for polynomials of n-limit length then the set is a Vector Space.}[/tex]
(Wow that took a while to type up...)
Let the set of all integrable functions be call V. All integrable functions can be expressed as a Taylor Series, all integrable functions can be defined as Taylor Series, when can then be integrated.
Let [itex]\oplus[/itex] be an operation defined by:
Let u, v, w be integrable functions defined as a Taylor Series [itex]\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}[/itex]
[tex]
u = \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}\\
v = \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n}\\
w = \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n}\\
1. u \oplus v = v \oplus u: \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} = \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n}\\
\begin{align} 2. u \oplus (v \oplus w) = w \oplus (u \oplus w&): \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus (\sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n})\\ &= \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n} \oplus (\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n})\\ &= \sum_{n = 0}^{\infty} \frac{h^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} \oplus \sum_{n = 0}^{\infty} \frac{g^{n}(a)}{n!}(x-a)^{n} \end{align}\\
4. u \oplus -u = 0: \sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} + -\sum_{n = 0}^{\infty} \frac{f^{n}(a)}{n!}(x-a)^{n} = 0 \\
\text{Closed Under Addition} \oplus \text{, closed as:} \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}}\\
\text{Let } \odot \text{ be an operator defined by:} \mathbf{c} \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\ \text{ where } \mathbf{c} \text{ is a constant }
\forall u, v\epsilon \mathbf{V} \text{ and c,d}\epsilon\mathbf{R}\\
\mathbf{1) } \mathbf{ c} \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{c}\\
\mathbf{2) } \mathbf{ c } \odot (\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \sum_{\text{n=0}}^{\infty} \frac{g^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}) = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{g^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\mathbf{3) } \mathbf(c+d) \odot \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} + \mathbf{d}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\mathbf{4) } \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{c}) \odot \mathbf{d} = \mathbf{c} \odot (\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} \odot \mathbf{d})\\
\mathbf{5) } 1 \times \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}} = \sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\\
\text{Scalar Multiplication } \odot \text{ closed as: } \mathbf{c } \odot \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}} = \int{\mathbf{c}\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}} = \mathbf{c} \int{\sum_{\text{n=0}}^{\infty} \frac{f^{\text{n}}\text{(a)}}{\text{n!}}\text{(x-a)}^{\text{n}}\text{dx}}\\ \\
\text{So, if the properties of } \oplus \text{ and } \odot \text{ hold under V for polynomials of n-limit length then the set is a Vector Space.}[/tex]
(Wow that took a while to type up...)
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